CHAP. I. Of the parts of the Planisphear. And of the Mater, his matter and Lineaments.

THis Planisphere is made up of five parts

• 1 The lying plate, called Mater.
• 2 The moving plate; called the Rete, Reet, or Net.
• 3 The Ring or Limb.
• 4 The Label.
• 5 The Sights.

The Mater is a round plate of metal or past-board flat smooth, and stiff: the larger the better. And if you will have the cir­cles actually drawn for every degree, it had need be ten Inches at least in Diameter. If it be made of metal (as Silver Brass or Tin and Tin-glass in equal quantity melted together) it must be well pollished: but it may very well be made on a fair past­board, pasted on a Messie board: for thereon the Lineaments may be distinguished with inkes of several colours, which cannot be if it be made in metall.

The Lineaments of the Mater (though they be fitted to re­present [Page 2]other circles of the Sphear also, as shall be shewed, yet) most aptly represent the Meridians and Parallels: and there­fore so we call them here, while wee speak of the Fabrique. Among the Parallels the two Tropiques and the two Poler cir­cles are to be inserted. And lastly to these you shall add the Ecliptique: and so you have all the Lineaments of the Mater.

For the Declination whereof,

1 Upon the center of the Mater plate, describe the funda­mental circle, (about an inch within the edge, if your plate be 12 inches Diameter) that so a convenient space may be left for the Limb. This circle shall be the great Meridian passing through the Poles of the world, and also through the Zenith and Nadir of your Countrey; and is the bound by which all the Line­aments of the Mater are inclosed. Draw two Diameters crossing one another in the center at right angles, and dividing this Meri­dian into his quarters, let one of these Diameters be A B the Axtree-line, the other C D the Equator or his Diameter: divide also every quarter of this Meridian into 90 equal parts or degrees.

This Meridian onely of all the circles of the Mater, falleth out to be a full circle in this projection, because the bisection of the Sphear is supposed here to be made in the plain thereof, and be­cause the eye is supposed to be the Pole thereof, and so equidi­stant from every part thereof. The rest of the Meridians and Pa­rallels their Semicircles are in this Planisphere fore-shortned ac­cording to Optique reason, (as shall be further explained Ch 2) because they all either are great circles passing through the eye­point, and cutting the Meridian at Right angles, as the Axtree-line or East Meridian, the Equator, and the Ecliptique, which are therefore projected upon their Diameters and become straight lines; or else lie Oblique to the eye, as do all the rest of the Meri­dians and Parallels, which are all of them projected into arches less then a semicircle; and yet every one of them is to be accoun­ted a semicircle fore shortned, and to be divided as a somicircle into 180 degrees.

2 To describe the rest of the Meridians, you shall lay a Ru­ler from the Pole A to the several degrees of the semicircle C B D (or from the Pole B to the several degrees of the semicir­cle [Page 3]C A D, for all comes to one) and mark where the Ruler intersects the Equinoctial line C D for every degree; so shall you have the Equinoctial divided into 180 deg. ( viz. 90 degr. on one side the center, and 90 degr. on the other) through which divisions or degrees the severall Meridians shall pass, and all of them must meet in both the Poles. Therefore having three points given for every Meridian ( viz. the two Poles A and B, and a middle point in the Equator C D, bring these three points into a circle, and you shall have the true arch of every Meridian drawn on the Planis­phear; as they appear in this Projection: which how to do shall be further shewed Chap. 3.

3 To describe the Parallels, you shall first divide the Axtree-line A B into 180 deg. (as you divide the Equator C D) by laying your Ruler from C or D to the several degrees of the op­posite Semicircle, and marking where the Ruler intersects the Axtree-line A B; or with your Compasses transfer the division of the Diameter C D to the Diameter A B; so shall you have three points given for every Parallel, whereby his arch may be drawn: as for example, the tenth Parallel North shall be drawn from the tenth degr. of the Quadrant D A, counted from D, by the tenth division of the Semidiameter E A, counted from E, to the tenth degr. of the Quadrant C A, counted from C: and so of the rest. The Tropiques and Poler circles are described as the Parallels. Yet of them see Chap. 6.

4 To describe the Ecliptick, number his greatest Declination (23 degr. 30 mi.) in the Meridian from C, towards A to F. and again from D toward B to G; then joyn the points F and G with a straight line which shall pass through the center of the Mater and be the Ecliptick. For dividing whereof you shall on­ly transfer the division of the Axtree-line or Equator upon this E­cliptick line (for all Diameters have like divisions) and you shall distinguish every tenth and 30 degr. by longer stroaks, and shall set ♑ at G; ♈ and ♎ at the center; ♋ at F: and the rest in their order.

CHAP. II. Of the reason of this Declination.

FOr the better understanding of the reason of this Declinati­on, either take or suppose to your self an hollow Sphear or [Page 4]Globe, of equal diameter with your great Meridian before drawn. Cut this Sphear by the Meridian into two dishes or Hemisphears, the one representing the Eastern, the other the Western Hemis­phear of the Heaven: So in each of these Hemisphears the Meri­dian is an whole Circle, being the Base of the Hemisphear, or brim of the dish: but the rest of the Meridians, the Parallels, Equi­noctial, and Ecliptick, are all bisected by the great Meridian, and so you can have but half their circles in the Eastern Hemisphear, and the other half is in the Western.

Next suppose also the plain A C B D, to be a thin plate of some transparent matter, as clear horn glass, or christal; and this plate fitted to stick in the dish mouth (that is, in the Meridian of the Hemisphear) and upon the center E a straight wyre to be e­rected perpendicular to the plain A C B D, and the length of the said wyre to be equal to the semidiameter E A. Now place your eye at the top of this wyre and look up the Lineaments of the He­misphear through the glass plate, and observe where the visual lines drawn from the severall points of the Hemisphear to the eye cut the glass and what kind of lines and arches they do paint up­on the glass, and you shall see there that the semicircles of the E­quator, the East Meridian, and the Ecliptick, will be depainted on the glass in straight lines; because they be great Circles, and pass through the eye point of this projection) and the lines passing be­tween your eye and the several degrees of any of them shall di­vide their Diameters upon the glass into such parts as they are divided by the precept of the former Chapter. For the top of the wyre (here supposed to be the West point of the Horizon) is the true eye point from which the Diameters A B and C D are divided, and from which the whole projection of the Eastern Hemisphear is made at one view upon the glass But because this concave Hemisphear with his wire or Axis erect, cannot ea­sily be pictured on a flat, therefore to supply the want of a solid Sche m you may consider that the Axtree line A B is not onely the Diameter of the East Meridian, but a common Diameter to all the Meridians: and therefore if you take your ey point at C or D, and by your eye beam or a ruler laid from one of those points to all the degrees of the opposite semicircle do divide the Axtree line, it shall be all one as if you had divided him from the top of the wire by the degrees of the East Meridian which passeth through the bottom of the concave Hemisphear. For all great [Page 5]Circles of the Sphear and their Diameters likewise being equal, look how any one of their Diameters is divided, in like man­ner they shall all be divided, if the eye be alike situate to them.

CHAP. III. How to find the centers of the Meridians five several wayes.

THe centers of the Meridians and their semidiameters are thus found,

• 1. You have by the first Chap. three points given by which they must be drawn (viz.) the two Poles, and a third point in the Equator) and how to bring these into a circle or arch, is shewed by Euclid. 4, 5. But I think this way incomodious for this purpose.
• 2 A better way is to get the centers by proffers, thus,

Let it be required to draw the twentieth Meridian whose points given are A 2 B. Having first extended infinitely the line C D, set one foot of your Compasses at adventure, in the line C D, whereabout you guess the center should be, and extending the other foot to A or B, carry it about at the same extent towards 2, and if it touch the point 2, you have taken the true center, and may draw the arch as is required, if your Compass over­reach, you must narrow it, and come nearer; if it reach short, widen your Compass, and seek your center further, till by tryal you light upon it: if one foot of your Compass stand in the line C D, and the other cut the middle point, and one of the extream points A or B, it ought to cut the other also; if your plain be flat, and the line C D straight and square to the line A B: but if any of these have failed, you shall never bring the three points into an arch while the foot of your compass [...]andeth in the line C D. Therefore in such case set one foot in A, and draw with the other foot a short arch crossing the line C D; then set the standing foot in B, and with the running foot cross the short arch last drawn: where these arches cross will be the center, by which you may draw an arch cutting A and B, and if it cut 2 also, you have your desire. But if this arch over reach 2, widen your Compass: if it [Page 6]come short of 2 (the middle point) narrow your Compass, and try again in like manner till you can compass all the three points in the same arch.

3. A third way. I suppose you may know that every Meridian cuts the Equator twice, viz. in two opposite points distant 180 degr. one from another: as, for example, the Meridian which cut­teth the Equator in 60 degr. of Right Ascension, cuts it again in the opposite degr, viz. 240. Now if you can find these two points in the Equator line C D, the center will be in the just middle betwixt them. One of these points is already given within the fundamental Circle; the other without, is thus found. Prolong your Equator infinitely beyond your plate both wayes, and divide the extension thereof by like reason as you divided his Diameter, viz. as by a Ruler laid from A to the several deg. of the Quadrant B C, you devided the Semidiameter E C into 90 degr. so keeping still one end of your Ruler fixed at A, and car­rying about the other end thereof to the severall degrees of the Quadrant C A, you may divide the excurrence of the line E C into 90 degr. more: and so E C and his exccurrence or continua­tion will be half the Equator divided into his degr. and E D with his excurrence on the other side will be the other half divi­ded by like reason. And thus the whole Equator is projected in one straight line, and divided into degr. also. Then having a point given within your fundamental circle through which the sixtieth Meridian must pass ( viz the 60 deg. of the Semidimeter E C or E D) number thence over the center to 180 deg. and there is the point where the other semicircle must cross, and the middle be­tween those points is the center. But because the two points taken in the quadrant A C are very near together, especially to­wards A, and the Ruler also will cross the prolongation of the line E C very obliquely, you may therefore do better to divide this line into his degr. by a Scale of Tangents, for if upon the Equi­noctial line D E C you prick down from E both wayes the Tan­gents of the half degr. in order from 0, to 90, those pricks shall be the whole degr. of the Equinoctial line in this Projection, to be numbred from E both wayes to 180 deg. where the Tangent be­comes infinite. Thus taking A E for Radius, E D is the Tangent of 45 deg. by the structures: yet the arch or Diameter E D is a Quadrant or 90 deg. of the Equinoctial in this Projection, because [Page 7]the Tangents of the half deg. of the Quadrant E A I, measure out the whole degr. of the line E D, as was above-said.

4 If you consider well what hath been said, you will find (or you may take it here upon trust,) that for the 90 Meridians to be drawn between C and E, half the centers will be found in the opposite Semidiameter E D, and the other half without D in the said Semidiameter prolonged. And that every second di­vision of the line E D from E toward D and forwards, shall be the centers of the Meridians which cut the Semidiameter C E. As for example, to draw every fifth Meridian from C to E, you take every tenth deg. from E toward D for the centers.

And further if you would not be at more trouble then needs, to divide the extension of the Diameter beyond the fundamental circle, you shall but do thus. Begin with the crookedest Meri­dians first, whose centers are within the fundamental Circle, and first pitching one foot of your Compass in the point 1, (near E) extend the other foot beyond the center to 2; there is the center from which you shall draw the first Meridian A 1 B: and also turning about your Compass you shall make a marke in the ex­tension beyond D at 1, where the other Semicircle of this Meridi­an would cross the Equator. So for the next Meridian, in the line C E marked with 2, your center is beyond E at 4, and after you have drawn his arch A 2 B, marke with your compass his other crossing at 2 beyond D, and so with one labour you shall both draw the 45 crookeder Meridians, and also make the out lying division of the line E D prolonged: of which division every se­cond or even number will be a center to some of the straighter Meridians. This is a very good and easie way: and this way Mr Blagrave alwaies used.

5 Or lastly You may frame a decimal Scale of 1000 or 10000 parts, equal to the Semidiameter of the Mater; by which Scale with the help of the Cannon of Triangles, you may presently find the length of any S ne Tangent or Secant you shall desire. Now look what inclination any Meridian hath to your fundamental Circle (that is, what angle they make between them) the Secant of that inclination is the Semidiameter of that Meridian; and the Tangent of the same inclination is the distance of his center from E the center of the Mater. For example, the Meridian A 2 B his inclination is 20 deg (for the angle C A 2 and likewise the arch C 2 which measures it is 20 degr. the S cant of 20 degr is [Page 8]10641. by the Cannon of Triangles (which every Mathematician ought to have at hand) Take with your Compasses from your decimal Scale 10641. and setting one foot in A, with the other foot cross the Semidiameter E D; in that crossing is the center: or take with your Compasses 3639. the Tangent of 20 degr. and set it from E toward D, and it shall give you the same center at 4. For A E being Radius, E 4 is the Tangent, and A 4 the Secant of 20 deg. by the structure.

And if you like to work this way, it will help you much to have a short Cannon of Tangents and Secants of whole degr. of the Quadrant gathered into one page: which Cannon for your ease is here annexed.

CHAP. IIII. To find the Centers of the Parallels, six several wayes.

THe first way, but the worst for our purpose, (as was said before for the Meridians) is by the fifth Propositi­on of the fourth book of Euclid; to find the Center of the Circle circumscribing the Triangle made by the three points given.

2 A better way is by profers. Take this upon trust: that as you found the Centers of all the Meridians in the Equator, so shall you find the Centers of all the Parallels in the Axtree line prolonged, and by making like profers as you were taught for the Centers of the Meridians, (Chap. 3.) you may quickly find the Centers of the Parallels.

3 A third way. You must consider that the Axtree line repre­sents the East Meridian as well as the Axis of the world which is a common Diameter to all the Meridians. Also that every Parallel cuts the East Meridian (as it doth the rest) in two points Equidistant from the Equinoctial and two Equidistant also from the Poles. Therefore having one point already given in the Axtree line within the fundamental Circle where the Parallel shall cut, number the distance from this point to the next Pole, and num­ber also the same distance again beyond the Pole in the Axtree-line prolonged, (being divided also as you were taught to divide the Equator line Chap. 3. and at the end of this number shall the Parallel out the Axtree line again. And the middle between these two sections is the Center. For example, the 50th Parallel is 40 degr. distant from the Pole. Count therefore in the Axtree line prolonged 40 degr. beyond the Pole, and there is the utter end of this Parallels Diameter; which if you part in two, the middle at G is the Center.

4 If from the point given, where the Parallel cuts the great Meridian, you raise a Tangent line, this Tangent shall cut the Axtree line in the Center of the Parallel. Example, The said 50th Parallel cuts the great Meridian at H, there I raise the Targent H G perpendicular to the Radius E H. And this Tan­gent as you see cuts the Axtree line in G the Center of the Parallel.

5 Hence ariseth a fifth way. For it appears by this figure [Page 10]that the Tangent of the Parallels distance from the Pole is equal to his Semidiameter: and that the Secant of his distance from the Pole is equal to the distance of his Center, from the Center of the great Meridian. For here E H is Radius, H B an arch of 40 degr. H G the Tangent thereof, and Semidiameter of the Paral­lel, E G the Secant thereof, and the distance of the Center of the Parallel from the Center of the Meridian. And all this is evi­dent by the structure in the Scheam. Wherefore making E H Ra­dius, take from your Scale or Sector, with your Compasses the Secant of the Parallels distance from the Pole, and set it from E in the Axtree line, and it shall end in the Center of the Parallel. Or take the Tangent of the Parallels distance from the Pole, and set it from the point of his Section with the Meridian toward the extension of the Axtree line, and where the end of it just toucheth the Axtree line, there is the Center.

6 For want of a Sector, or other fit Scales of Tangents and Secants you may do thus: Set one foot of your Compass in the Center E, and extend the other upon the Diameter of the Equa­tor or Axtree line, to twice so many degr. as your Parallel is distant from the Pole. That distance is the very Tangent you seek. For example, for the 40th Parallel from the Pole, I number from E toward D 80 degr. to 8. now E 8 is the Tangent of 40 degr. (though it contain just twice so many degr. of the Circle foreshortned in this projection: as hath been shewed Chap. 3. Sect. 3.) and so if you will have the Secant of 40 degr. take with your Compasses the length from 8 (where the Tangent ends) to A. and that is the Secant to be used as was taught in the last Section.

Thus have you wayes enough for finding the Centers of the Me­ridians and Parallels. And you may have occasion in the making of the Instrument to use most of them one time or other. How­ever, the knowledge of them is both pleasant, and usefull for the right understanding of this and other Projections of the Sphear, as also for the examination of your work, when you shall chance to doubt of it.

CHAP. V, How to draw the straighter Meridians and Parallels, whose Semidiameters are very long.

IT may trouble you very much to draw those Meridians and Parallels which lie near to the Diameters; because they be arches of great Circles and require Compasses larger then you can well get, or manage when you have gotten them. Till you come to the 80th Meridian from the Limb, a Beam­compass of a yard long will reach, if your Mater be not above a foot Diameter, and a longer Beam you cannot well manage, for it will be apt to tremble with it's own weight, and draw double lines, though it be made very thick and massie. But the 89th Meridian will require a Beam-compass of almost ten yards long: For his Semidiameter will contain the Semidiameter of the great Meridian 57 times. Therfore to draw the 10 last Me­ridians and the 10 last Parallels, you may help your self one of these wayes.

• 1. Guido Ʋbaldus hath devised an Instrument for this purpose, consisting of three rulers in form of an obtuse Triangle. The de­scription and use thereof you may see in Blagr. l. 4. c 2, 3. and in Ʋbaldus his book De Theorica Astrolabij. But though it be an Ingenious device, yet I have found by experience, that it is a ticklish Instrument, and hardly managed, for which reason I have hanged it by.
• 2 The Bow now commonly used, is an Instrument not so ar­tificial, but more tractable and steddy then the former. It is made of too steel rulers, the shorter of them must be of good sub­stance, as three quarters of an inch in heighth, and as much in breadth, that it may be stiff, and lie flat; the length must be some­what more then the Diameter of your Instrument: The other may be an inch longer, of the same heighth but much narrower, that it may be bent out with screws into an arch of any Circle required; which ruler so bent, being laid to the three points given, you may by it draw the arch required, as easily as you draw a straight line by a straight ruler, The stiff ruler carries the screws, and it must have rivets, by which the bending ruler may be staied at both ends, while it is bent by the screws. See the figure.

CHAP. VI. How to draw the Tropiques, and Polar Circles, and to finish the Mater.

BEsides the 180 Parallels aboy ementioned, you have four more to draw before the Mater is finished. viz. the two Tropiques, and the two Polar Circles; of which the Nor­thern is called the Arctique, and the other the Antarctique Cir­cle. How to draw these you are sufficiently instructed Chap. 4. if you know but their Declination, for they be Parallels. The Tro­pique of Cancer declineth from the Equator toward the North Pole 23 degr. 30 min. and the Tropique of Capricorn declines as much toward the South Pole. The Arctique Circle declines Northward 66 degr. 30 min. and the Antarctique as much Southward. And these being drawn after the manner of the o­ther Parallels, you have drawn all the lineaments of the Mater, And the better to adorn and distinguish them you shall with your Graver hatch every fifteenth Meridian: for they are hour lines. The South arch of the great Meridian A C B is the hour of Noon: and his North arch A D B, the hour of Midnight. These need not be hatched, being the Semicircles of the great Meridian, or fundamental Circle, which contains all but the Axtree line A E B, which is the hour line of the sixes, and the rest of the hour lines counted from him both waies, would be hatched on both fides, to shew like a ragged staff, for distinction sake. Also every fifth Meridian (not being a fifteenth) you shall make a pricked line, not punching it with a round point, lest you make your plate warp, but making many short strokes cross the line with your Graver, which will be more conspicuous. Every tenth Parallel also would be a ragged line, and the intermediate fifths pricked lines: likewise the Tropiques and Polar Circles would be pricked lines. Also if your plate be large, you may set figures to the hour lines, and to every tenth Meridian at the Equator: but if your plate be smal the divisions of the Label ap­plied upon the Equator may supply the lack of them.

CHAP. VII. Of the Reet, or Nets

HAving shewed you what belongs to the Fabrique of the first part of this Instrument called the Mater: A few words more will instruct you how to make the Reet, whose lineaments are for the most part the same.

The Reet is a round plate of metal, or pastboard, like unto the Mater, but of less Diameter: it must be well planished and poli­shed; and the thinner the better, if it hold working it would not be thicker then a shilling, being of a foot Diameter. It is called the Reet or Rete, that is the Net, because it must be pierced through, and made like unto a Net or Lettess; that the lincaments of the Mater may be perceived through it. If we had a trans­parent metall, much labour might here be saved. A clear Lan­thorn horn may serve for a smal Instrument, but for large Instru­ments, it is best to have it either of fine pastboard, or, if you will go to the cost, of metal cancelled; as shall be taught.

1 For the delineation of the Reet, first draw your fundament­tall Circle equal to the fundamental Circle of the Mater leaving a border or Limb without, of such breadth as may receive the graduations of the Circle and figures set to them, which breadth may be three tenths of an inch, where the Reet is a foot in Dia­meter: draw likewise two Diameters A B, and C D, crossing one another in the Center E at right Angles, and dividing the Circle into his four Quadrants, which you shall subdivide again into 90 degr. apeece, as you did in the Mater.

2 You shall inscribe two arches, which shall represent the Se­micircles of the Ecliptique, which shall meet at the points C and D of the Equator, and the middle points of these arches shall be found in the Diameter A B, thus. The Diameter A B being di­vided as before you were taught to divide the Diameters of the Mater, number from A toward the center E 23 degr. 30 min. and there make the point F, for ♑: and likewise number from B toward E 23 degr. 30 min. and there set the point K. for ♋. then join the points C F D in one arch, and the points C K D in ano­ther arch (as is taught Chap. 3) and your Ecliptique is drawn. But now you must make him a narrow Limb inward toward the [Page 14]Center, to receive the scale of his degr. and the characters of the Signes. And to divide him you shall do thus. Number in the Axtree line (A B, from F inwards 90 degr. there is the Pole of the arch C F D, to this Pole fasten one end of your ruler (having an ey-lid-hole in the edge for that purpose) and carrying about the other end over the several degr. of the Semicircle C A D, you shall cut the arch C F D into his correspondent degrees. As if you lay the ruler from C to 10 degr. in the Limb toward A, it shall cut the Ecliptique in ♎ 10, and so of the rest. Likewise for the other Semicircle C K D, find his Pole 90 degr. from K to­ward F and A: and from that Pole by like reason, you shall di­vide the Semicircle C K D by the divisions of the Semicircle C B D. This is the best way.

Or you way divide the Ecliptique by a Table of Right As­censions, thus. Lay your ruler from the Center E to 27 deg. 54 min. in the Limb, which is the Right Ascension of ♉ 0. to be counted from D towards B, and the ruler shall at the same time cut the Ecliptique in ♉ 0, to which that Right Ascension belongs, and so for any other deg. or you may defer the dividing of the E­cliptique, till you have finished and cut out the Reet: and then if you set the line C D of the Reet, in A B the Axtree line of the Mater, the Ecliptique will lie among the Meridians of the Ma­ter, and shall be so divided by the Parallels of the Mater, as the Meridians are divided by them. But my advice is that you di­vide your Ecliptique the first way, and you may use this for proof of your work at last.

3 The rest of the lineaments of the Reet are the Azimuths, to be drawn as the Meridians of the Mater, and the Almicanters, to be drawn as the Parallels. Onely you shall need to draw but half the Almicanters, and the Azimuths but half way, leaving one half of your Reet viz. E C B D blank and void of them.

In drawing these Azimuths and Almicanters you shall be carefull to skip over the border of the Ecliptique, leaving it fair, that the graduations thereof with their figures set to every tenth degr. and the characters of the Signes may be more distinctly seen. Also you shall do well, if you make a border to the Axtree line on the Northside, that is toward D, and let this border be of the same breadth from A to B, the breadth not exceeding one fifth of an inch in a Reet of a foot Diameter; upon which border you may make a scale of degrees, setting figuresin it to every tenth [Page 15] Almicanter. This will be a great strength and Ornament to your Reet.

Below the Horizon C D likewise, you shall make a Limb or border for the Horizon, to receive his graduations: this may be a quarter or three tenths of an inch broad, where the Reet is a foot in Diameter: and upon this border you shall set figures at eve­ry tenth Azimuths, and shall number them both wayes, from the Center and from the Meridian.

4 You shall inscribe so many of the fixed Stars as your Reet may well receive. Which to do you must know their Right As­censions (or Culminations) and also their Declinations, for which purpose I have given you a Table of 110 of the more notable fixed Stars, (which may best be inserted in your Reet,) with their Right Ascensions and Declinations calculated to the year of our Lord 1671. which may serve for 40 years before and after with­out any considerable error.

To inscribe them you shall first number the Right Ascension of the Star from ♈ 0, that is from D upon the Limb of the Reet toward B, and at the end of that number fix your Label, (which by this time should be made and pinned on the Center,) then from the Limb count inwards upon the Label the Stars Declina­tion, and at the end of that number make a prick in your Reet close to the edge of the Label, there is the Stars place. Then with your Graver you shall make there the shape of a Star, with 4, 5, or 6 points, according as the magnitude of the Star deserves: and let one point be longer then the rest; and let it point outward from the Center, if the Stars Declination be North, but inward toward the Center, if his Declination be South: and let the end of his long point (called Apex) be in the very true place of the Star. But if your Label be yet unmade, then take the measure of the Stars Declination with your Compasses upon any of the four Semidiameters of the Reet, (measuring it from the Limb in­wards,) then lay a ruler from the Center to the Right Ascention of the Star and where the ruler cuts the Limb of the Reet; there set one foot of your Compasses, (opened as before) and with the other make a prick toward the Center, close to the edge of your ruler, and there is the Stars place in your Reet.

5 Lastly, you shall cut out all the spaces of this Reet, which may be spared; remembring alwaies that you leave uncut the borders of the Ecliptique, Horizon, and Axtree line, and be very [Page 16]carefull that you cut not into the Center of your Reet, but leave breadth sufficient about the Center to hold the Center-pin, which must joyn Mater Reet and Label together. This remem­bred, you shall cut out two third parts of the spaces of the Al­micanters, beginning from the Horizon C D, and cutting out the breadth of two degrees, after which you shall leave the breadth of one degree; and then cut our again the breadth of two degrees, and so forward. But for the greater strength and ornament of your Reet, and for ease in numbring the Azimuths, you shall at every 15th Azimuth leave a string of the breadth of one degree, whole from the Horizon to the Pole A, or at every thirtieth Azi­muth leave such a string going quite through, and at every other fifteenth the string may be cut off when it comes within ten degr. of the Pole, because there the spaces of the Azimuths be very narrow and close together.

And where among those Almicanters and Azimuths you have any Star, you must contrive to leave him standing, and to set by him his name or some figure by which you may know him again. But you are to content your self with four Stars on this side the Horizon, because you will want convenient room. On the other side you may have more, and room also to writ their names upon strings or branches left for that purpose; which you may contrive into some voluntary lettess-work, wherein you shall not much re­gard uniformity of the Quadrants, but to make the Reet as open as you can, provided you leave it of sufficient strength.

The cutting of this Network requires much labour and care. be sure you use no Punches nor Ch [...]sils, nor adventure to stamp your figures, lest you spoile all. But get your Gravers Drils and Files, of several bignesses and fashions, and being to cut out any of the lettesses, lay your plate upon a flat board, or contrive to pintch it between two flat boards, as in a Vice, that it crumple not. The lower board were best have a smal Auger hole, through which your Files may play: the upper board must come close to your work behind, but must not cover it. Then first make way for your Drill with a Graver: next with your Drill (set Perpen­dicular to the plate) bore two or three holes close together, then get in a smal File, and file them all into one mortess or narrow window, to make way for bigger Files, and when you come near the line be sure your File be not slope but Perpendicular to the plate in working. And to guide your Files Perpendicular in [Page]

[Page 17]working you may make a long handle to your File, like an arrow shaft, and longer if you will: fasten a board with an hole befitting the handle of your File just over your work; put the shaft or handle of your File through that hole so far that it slip not out in working: and this hole shall so govern your File, that if you set it Perpendicular at first it must needs file Perpendicularly. This device I learnt from an excellent Artificer, M. Matthew Hill, of Bedford; who by this and such other ingenious devices did cut out two of these Reets so exactly and truely, as I think the like hath not been done in metall before.

To cut out the Reet in pastbord is much easier, if you be pro­vided of sharp knives and chesills, fitted for your purpose.

A Table of the Right Ascensions and Declinations of 110 of the more notable fixed Stars; calculated from Tycho his Tables: rectified for the year of our Lord 1671.

CHAP. VIII. Of the Ring, or Limb of the Mater.

THe third part of this Instrument is the Ring or Limb, which is nothing else but the skirt of a Circular plate equal with the Mater, whose middle is cut out by a lesser concentrique Circle. It is bounded with two Pa­rallel Circles, the outmost must touch the edge of the Mater round the inner Circle or edge of this Ring: it must be a little less then the Limb of the Reet, that it may take hold of the Reet to keep it flat and safe from harms. This Ring had need be thicker then the Reet, but not so thick as the Mater; and for breadth, a­bout 1/10 part of the Diameter. It must be pinned or screwed on to the Mater with 6 or 8 pinnes or screws, that so you may take your plates asunder when need is to cleanse them from any stain or dust that may get between. Let the pins that carry the naile-screws be riverted in the Ring, and chair heads so filed down and polished, that they be not seen to check the Label: and holes being made in the Mater for the pins to pass through, you shall have smal screws of what fashion you like best, to turn upon them on the backside; these screws would be made all of a length, and may serve as feet to bare up the Instrument about a third part of an inch from the ground, that it be not scratched and be rea­dier to take up.

And that the Reet may turn more pleasantly under the Ring, and lie as near as may be in the same plain with the Ring, you shall abate half the thickness of the upper edge of the Reet, about a barly-corn's breadth round about, so far as he shall run under the Ring and likewise aba e half the thickness of the inner edge of the Ring on the lower side where he clasps down the Reet, (which a good Turner knowes now to do;) or you may make a shift to do it with a beam-compass, if you make your running point like a narrow chesill.

Your Ring thus fitted to the Mater, you shall set one foot of your Compasses in the Center of the Mater, and with the other draw near the inner edge of the Limb a Circle about 8/100 of an inch distant from the edge. Also opening your Compasses about ⅙ of an inch more you shall draw another Parallel Circle: and [Page 21]laying your Ruler or Label from the Center to the several de­grees of the fundamental Circle of the Mater, you shall draw short lines for every degree from the inner edge of the Ring to the first Circle, and every tenth degree you shall prolong to the second Circle, and let every fifth be drawn half way. Between these two Circles also you shall set figures to every tenth degree; numbring from the Equinoctial line C D to the Poles on both sides both wayes. Also without the second Circle you shall set great figures for the several Hours, setting XII at C, and thence proceeding in order to the right hand toward B, at 15 degrees set I, at 30 degr. set II, at 45 degr, set III, and so on, till you come to D, where you must set XII. Thence you shall proceed in the other Semicircle D A C, setting I, II, III, and so on in order, till you be come round. And remember that you write on the Ring, at A Oriens, at C Meridies: at B Occidens, and at D Septentrio.

CHAP. IX. Of the Ephemeris or Calender, on the Ring.

IF there be space enough left upon the Ring without the Circles of the degrees and Hours, you may fill it up with the Ephemeris of the Sun in this manner:

The former Scale on the inner edge of the Ring shall serve you to this purpose for an Ecliptique; and you may set to him the Characters of the Signes, if you will, at every thirtieth degree; beginning at Oriens and there setting ♈, and ♋ at Meridies, and the rest in like order.

Then draw another Scale without this, upon the Ring, con­sisting of two spaces. In the inner space shall be the Dayes of the Year: in the outer space: which must be a little larger) shall be the Names of the moneths in their order.

And to divide this Scale rightly, you shall do thus. Go to some Eshemeris for the Leap Year that next comes, viz. 1660, (or rather for some Leap Year about 20 Years hence, that your Scale may serve without any Prosthapheresis, for 40 Years to come without sensible error, and beginning your year with March, look where the Sun was on the 29 of February at Noon; which you shall find to be ♓ 20 degr. 47 min, for the Year 1660. There­fore laying the Label or a Ruler from the center to ♓ 20 degr. [Page 22]47 min. in the inner Scale, strike a long stroke through your out­ward Scale, and from thence begin your Year, writing from thence toward the right hand March 1660. Then lay the Label to ♓ 21 degr. 47 min, (which is the Suns place on the first day of March at noon the same year) and where it cuts the outer Scale, mark the first day of March, and so the rest in order. And to the first day of every moneth, you shall set his proper Letter which belongs to him in the Calender; as to the first of March you shall set D, and to the first of April G, &c. and when you have done December, you must take the Suns place for January and Febru­ary out of the next years Ephemeris, viz. 1661, and note that the space for the last day of the year ( Febr. 28) will fall out to be less by a fourth part then the rest, by reason that the Sun wants almost 6 hours to finish his Circle, which he finishes in dayes 365, 5 hours 48 minutes. And for this cause these Scales will serve you to find the Suns place at noon, for any day in a like year, that is every fourth year, accounted hence, either backwards or forwards; which year shall evermore be accounted to begin from Febr. 29. and may be accounted the first year after Leap year, because the Intercalation was February 25 next before. Then for the year next following, viz, 1661. (beginning March 1 and being second from the Bissextile or Leap year) these Scales shall give you the place of the Sun at six hours after noon, and the third year from Bissextile 1662 (beginning as before March 1) these Scales shall give you the Suns place 12 hours after noon, or the midnight following. And the fourth year 1663 being Bissextile, these Scales shew the place of the Sun at 18 hours after noon, the next year 1664, being the first after Bissextile, and beginning (as aforesaid) March 1, is the very same year for which your Scale is made and gon, for that year: your Scale shewes the Suns place at noon again.

But because the Julian years are bigger then the true Solar years by almost 12 mi. of time (that is, near a quarter of an hour) in which time the Sun moves 27 sec. 13 thirds 37 fourths, therefore when you have found the Suns place by the former Scale, any year after 1660, look how many years are passed since 1660 and so many times you must add 27 sec. 13 thirds 37 fourths, (that is almost half a minute) to the Suns place found: and for years past [...]ou must subtract as much, that you may find the Suns place exactly. This Prosthapheresis in 2 or 3 years is scarce considerable [Page 23]in an Instrument, but in 10 years there will be 4 minutes 32. seconds, and in 20 years 9. minutes 5. seconds, to be added after 1660. and as much to be subtracted in like number of years preceding the year 1660. to which this Scale is supposed to be framed.

This Ephemeris or Calender M. Blagrave would have on the back-side, where hee would also have a Ruler with Sights, to take the Altitude of the Sun or Stars. But this will be found incommodious in many respects, both in the framing, and in the using; and therefore I advise that nothing be set on the back-side but the Tables of the Prime, Epact, and Cycle of the Sun, thereby to find the age of the Moon, her Conjunctions and Oppositions, and the moveable Feasts for ever, Of which see Chap. 11.

CHAP. X. Of the Label and Sights.

THe Label is a Ruler slit in the midest, and the half of it cut away to the Head, where it is pinned to turn upon the Center, and reaching to the outside of the Limb. The Fiducial edge thereof, which pointeth upon the Center, must be graduated, like to the semidiameters of the Mater and Reet, into 90 degrees, to be numbred either inward or outward. The fashion of it may be understood by the figure without more words.

To this Label you may fit Sights, either fixed or moveable, as you like best, for observing Altitudes and Azimuths: but for taking. Azimuths you had need have one tall Sight, at least half as long as the Label: and then it had need be movea­ble, to take off at pleasure.

For taking the Altitude of the Sun, I have made a pair of moveable Sights, to slip up and down upon the edge of the Pla­nisphear; having on the backside springing plates of brass to pintch them close, and make them stick where you set them. These are commonly to be set at C and D the ends of the Equi­noctial line. At A in the Limb and in the Circle next unto the inner edge which boundeth the strokes of the severall de­grees, you shall drill a small hole, through which you may put [Page 24]a thred to hang a plummet on. The Sun then shining through the Sights placed at C and D, the plumb-line shall shew his Altitude in the semicircle B C A, you beginning to number from B, and observing where the plumb-line crosseth the Circle in which the hole for hanging the plumb-line was made. And here you must remember that because the plumb-line falleth not from the Center of the Planisphear, but from a point in the circum­ference about A, therefore the space of two degrees must be ta­ken but one degree, so that if the Plumb-line fall 20 degr. below B toward C, the Suns Altitude is 10, degrees as you may see demonstrated, Euclid. 3.20. and Pitisc. Trigonem. 1.53. And thus you may observe the Suns Altitude neer the Horizon, as exactly as by a Quadrant, whose semediameter were equal to the diameter of your Planisphear. But if the Altitude exceed 30 or 40. degrees then will the Plumb-line cut the limb too slope and have too much play to your trouble: For remedy whereof you shall remove the Sight at D towards A some degrees: as for example 60 degrees, by which means you shall abate the Suns Altitude 30 degrees, which 30 degrees must be added to the Altitude observed: as for example, the Sights are placed one at C, the other 60 degrees above D toward A, and the Plumb-line cuts 10 degrees from B towards C, I say, then is the Sun 5 degrees high and 30 degrees more, in all 35 degrees: in like manner you may place the Sight at any other number of degrees from D toward A, as you shall find most convenient for the present Altitude; remembring always that how many degrees soever you remove the Sight, half so many are to be added to the Altitude found. But if your Reet happen to run so far under the Limb, that you cannot make a center-hole for the Plumbet through the Limb and Mater, without hindring the Reets motion, then may you have a small plate of sheet brass, in fashion of an Arm or Tongue, in the point whereof you shall have a Center-hole drilled, and this plate shall be so joyned with a sluce or screw about the Limb near A, that the Center-hole made in this plate may lie close to the point where the Center should have been boared in the Limb in the line A B: and thus you may put it on, and take it off, at pleasure, that it hinder not the motion of the Reet or Label. Of the fashion of the Sights see more Booke 4.2.

CHAP. XI. Of the perpetual Calender, on the back-side.

ON the back-side of your Planisphear you may set the Calender following, which consisteth of three Tables gathered round. The longest would be set outtermost. The first is the Table of the Cycle of the Sun, that is of the Sundays Letter. This is here placed in the middle. It is a Cycle of 28. years, in which time the Dominical Letter runs all his changes, (caused by the one odd day above 52 weeks) in every Common year, and two odd days which run over the even weeks in the Leap years. To find the beginning of this Cycle, add to the year of our Lord 9. (because the first year of our Lord, as wee commonly acount, was the 10 of this Cycle) and divide the sum by 28. the remainder is the year of the Cycle running; and if nothing remain, then it is the 28. or last year. So you shall find that the Cycle now running shall end with the Julian year 1671 as in the Table; and shall begin again with the year of our Lord 1672. Thus may you renew the Table when it is expired; or make this very Table serve you for ever.

Example.

Enter this Table with the year of our Lord 1656. now run­ning; and you shaall find over against this year in the next space inwards 13. shewing you that it is the 13 year of the Suns Cycle; (so shall the 28 year forward viz. 1684 be again the 13th year of the Cycle next comming.)

In the next space within, you have the Dominical Letters F and E, (because it is Leap year) F shall be the Dominical Letter till you come past the Intercalary day, which is the day follow­ing the 24th of February: and for the rest of this year the Domi­nical Letter shall be E; (for the Letters always change back­wards) also you shall note here that the day inserted in every 4th year is not February 29, but February 25. for February 24. (being 6 Cal. Martij) is repeated again in the Leap year: and they write again February 25.6 Cal. Martij: and the 25th of February in the Leap year is marked with the same letter F, [Page 26]wherewith the 24th of February hapneth always to be marked. Hence the Leap year is called Annus Bissextilis: and note that by the Eclesiasticall Law S. Matthias day, which is February 24th in common years, in the Bissextile years is to be observed on February 25. Nevertheless in our Secular Law not the 24th and 25th of February. but the 28th and 29th in the Leap year are ordained to be one day in the account of Law: as by Statutum de Anno Bissextili, made in 21 year Henry 3. may appear.

The second Table is of the Cycle of the Moon, consisting of the Prime or Golden Number, and the Epact. This Table contains 19 years, which is the Annus Metonicus: in which space of time Meton an Astronomer about 430 years before Christ, observed the Moon to finish all her variations. So that every 19 year the mean conjunctions or changes should hap­pen upon the same days of the moneth, that they did happen upon 19 years before; onely an hour and half sooner. Yet be­eause every 19 years contain not the same number of Leap years, but sometime there come five Leap years in 19 years, and sometimes but four, therefore there may happen in this Period of Meton an error of an whole day, besides the hour and half above mentioned. For remedy whereof Calippus, about the 330 year before Christ, devised to quadruple this period of Meton, making Period [...]s Calippica of 76 years, which contains just 19 quaternions of years; so containing always the same number of Leap years and days. This period is therefore more perfect then Metons: for after this period of 76 years, the Moon runneth over the same course for her conjunctions and oppositions; changing in the same year of the period always, upon the same day of the moneth, save onely that shee changeth sooner by six hours in the latter period then in the former.

But the Church still retaineth the period of Meton, called the Prime, or Golden number; because it used to be set in Golden letters in the Kalender, in a certain artificiall order throughout every moneth, to guide you to the day of the Moons Priming or Changing: so may you find it in red letters, thus, Set to the Kalen­der printed in large folio with the Book of common Prayer, in the time of the late Queen Elizabeth. To find the year of the Prime, add to the year of our Lord 1, and divide the sum by 19. the remainder is the year of the Prime, or if nothing re­main it is the 19th or last year. Thus you may find the present [Page 27]year 1656, to be the fourth year of the Prime, and so you find it in the Table. The Cycle or Period now running, ends with 1671. and begins again with the year following.

The Epact or Concurrent is set against the Prime in the next space inwards: and finisheth his Cycle in the same time. It was devised to find more readily the day of the change, and age of the Moon. The way to find it is this. Multiply the Golden number serving for your year by 11, and divide the product by 30; the residue is the Epact for your year. Or having the Epact known for any year, you may make it from year to year, by ad­ding 11 to the Epact of the year foregoing, and casting away 30 when the sum exceeds 30. The reason whereof is, that the Moon changeth 12 times in 354. days, that is 11 days before the Sun hath gone his round: for which cause the changes must needs happen every year 11 days sooner. Observe here that the Prime changeth every year the first day of January; the Epact not till the first of March. The Dominical letter changeth upon the first of January going one letter backward yearly; and in every Leap year it changeth again on the Lords day next after February 24. The reason whereof is this: The Calen­der is marked throughout with the letters of the week A B C D E F G, and the same day of the year is always marked with the same letter, now if the year contained even weeks then would the Dominical letter be always one and the same: but because a common year contains 52 weeks and one day, there­fore the last day of the year must be marked with A, as the first was; the first day of the year following again is marked with A: now put case the last of December marked A, be Sunday in the year 1654. the day following viz January 1, is Munday, and yet marked A; and the first Sunday in 1655 must needs fall on January 7 marked with G: and so G became Dominical letter for 1655. as A was for 1654. and as those 2 days mark­ed with the same letter A, coming together at every years end cause a change of the Dominical letter, so in the Leap year the Intercalary day February the 25, being marked with F, al­ways as the 24th is, the coming together of those two dayes marked with one letter, causeth a second change of the Domini­cal letter for that year by like reason.

The third Table is a Table to find Easter for ever. This is placed outmost because it is the longest. It was very falsly [Page 28]ordered in M. Blagraves book: and so it is in Grostons Tables, from whence I suppose he transcribed it. I have set it right and straight, and taken what care I could that the Printer or Graver do not put the ranks into the same disorder in which I found them, both in Blagrave, and in Grostons Tables, printed 8 years before him.

CHAP. XII. Some cautions to be Observed in the making of the Instrument.

THough I have taught the making of the Mater first as being the base and principal part of the Instrument; yet I shall advise you first to draw and cut out the Reet, and fit the Label to it, leaving it sufficient length to reach to the out-side of the Mater. And then having your Ring ready fitted to screw on to the Mater you shall drill the centers of the Mater, Reet, and Label, with the same drill; and fasten them together with a Center-pin well fitted to the bore: this Center-pin must not be too big: let it be square at the head to stick in the backside of the Mater, that it turn not; and let the other end have a male screw, upon which you shall turn a female screw, to draw the plates together; so that they be neither too loose nor stick too hard. Then take a great needle, or such a round point of hardned steel, and bearing it up into some corner of the Reet close to the inside of his Limb, turn about both Reet and Needle upon the Mater, so that the Needle may trace out the great Meridian of the Mater, and so the princi­pall Circles of the Mater and Reet shall be sure to be Concen­trique, and to argree in all postures of the Reet: whereas if you draw the Meridian of the Mater before you have bored the Center, and fitted on the Reet, you shall hardly happen to bore so true, but you shall find the Circles to be a little Eccentrique and interfering one with another. And to avoid the like Eccen­tricity in the Circles of the Ring, it were best to have a sorry Beam Compass made onely for this purpose, and fitted to your Center pin to draw them by: yet if those be drawn from the Center of the Mater unbored, there will be no perceiveable error, if you devide them from the Limb of the Reet by the Label when all is pinned together. These things done, quarter the Meridian of the Mater with two cross Diameters, and divide them as is above directed; remembring to apply your Reet to the Mater often, to see how the divisions of the Diameter agree, and how the Meridians and Parallels which you are drawing on the Mater agree with the Azimuths and Almicanters of the Reet. And thus by comparing your plates often, and examining the lineaments of the one by the other, as you draw them, you shall avoid many slips and mistakes, and proceed in your work with more confidence and contentment.

CHAP. I. Of the Planisphear in the Meridional Projection, repre­senting the Eastern or Western Hemisphears: And of his three Modes or postures.

WHat M. Blagrave Book 2. Chap, 15.16. calleth the first and second distinction of the Jewel, I call the first and second Meridional and Equinoctial Projection of the Planisphear, for which change of termes I hope the Judicious Reader will not blame me. The Meridional Projection is the Eastern or Western Hemisphear [Page 31]projected upon the plain of the Meridian of your place, which is the great and chief Meridian in every Country, passing through the Zenith and Nadir of the place, as well as through the Poles of the World. The eye in this Projection is supposed to be placed in the East or West point of the Horizon. The lineaments which belong to this Projection are the innermost Scale of the Ring, and all the lineaments of the Mater and Reet, except only the Zodiack of the Reet, and the Stars. Here the outmost Circle both of the Mater and Reet represents the great Meridian of your place, and the Scale upon the inside of the Ring divided into 360 degrees serveth to divide the said Meridian: for the Label laid upon any degree of the Scale of the Ring cutteth the same degree in the Meridian Circle; because it is concentrick thereto.

This Meridian also standeth for Colurus Solstitiorum, (so they call that Meridian which passeth through the beginning of Cancer and Capricorn) for though all the Meridians in 24. hours space do successively come into the Meridian of your place (which is the Noon Circle passing over your head North and South) and the Sphear may be divided into Eastern and Western Hemis­phear by any of these Meridians, when they become Vertical, yet the Sphear is then in the best posture to be divided for our purpose into East and West, when Cancer is Southing, Capricorn at midnight, and ♎ 0 rising full East, and ♈ 0 setting full West.

In this Meridian at A, or Oriens is the North Pole, and the South Pole at B, or Occidens: for the words Oriens and Occi­dens are there placed to serve the Equinoctial Projection. The Concurrent Circles meeting in the Poles A and B are Meri­dians. Those Meridians are 180 in number, and divide the Equator C D into 360. degrees, because, every one of them cutteth it twice, that is once in each Hemisphear. By these are numbred the Right Ascensions of the Stars and Planets, and the hours and minutes of Day and Night: for every 15 of these Meridians numbred from the Limb is an hour Circle, as hath been shewed (Book 1.6.) they are numbred from D to C, that is, from Septentrio to Meridies 1.2.3. &c. for the Morning hours and back again from C to D, in like manner for the Af­ternoon: the Axeltree line A B falling out to be the six a clock line both ways. By those Meridians also are numbred the Longi­tudes of Towns and Countries in Geography.

The Circles or Semicircles crossing these Meridians are the [Page 32]Parallels of Declination: they are lesser Circles whose propertie it is to divide the Sphear into unequal parts. In the midst of them lies the Equator C D, being here a straight line, and cut­ting the Axtree-line A B at Right Angles in the Center E: the Parallels are greatest near the Equator, and from thence they lessen toward the Poles, they are 180 in number i. e. 90 on each side the Equator, save that the two extream Parallels are redu­ced to two points in the Poles. By these Parallels are numbred the Declinations of the Stars in Astronomie, and the Latitudes of Towns and Countries in Geography.

And this name and use have the Circles of the Mater always in the Meridional Projection. The Ecliptick always standeth for it self, when it is used, which is onely in the first Mode of this Projection. But the Circles of the Reet have divers names and uses, in the divers Modes of this Projection, which here follow.

CHAP. II: Of the Equinoctial Projection: shewing the Northern or Southern Hemisphears.

THe Equinoctial Projection representeth the Northern or Southern Hemisphear projected upon the plain of the Equator. Here the Limb or outmost Circles of the Mater and Reet are Equator. The eye-point [Page 35]is the North or South Pole, which you will, by turns. Which Poles are here expressed on the Center of the Equator, because the Sphear is pictured on a plain or flat. The Axtree line of the Mater A B is Colurus Equinoctiorum, the Diameter C D crossing him is Colurus Solstitiorum. But contrary on the Reet the Axletree is Colurus Solstitiorum and the Finiter Colurus Equinoctiorum. The Colurus Solstitiorum on the Mater is also the Meridian of your place, and therefore is marked with Septentrio, and Meridies; and the ends of the Axtree with Oriens, and Occidens. The rest of the Meridians being all straight lines meeting in the Poles or Center, are casily supplyed by the Label: and so may the Parallels also, being Concentrick with the Equator. For if you lay the Label on the 15. degree in the Limb from Meridies toward Occidens, the fiduciall edge of the Label there designeth the 15 Meridian, or the One a clock line: the North Quadrant of the said Meridian proceeding from the Center (now the North Pole) outward to the Limb or E­quinoctial, and the South Quadrant returning in the same line from the Equinoctial to the Center: (now the South Pole:) and if you remove the Label 180 degrees from One a clock of the day there it shall designe One a clock at night, made by the other Semicircle of the same Meridian, which joyneth with his match in the Center without any angle, that is, into the same straight line: and so of the rest.

And for the Parallels, if you set the point of your Compass or a needles point in the 23. degree ½ of the Label, and turn about the Label with the point, it shall describe a Circle which will serve for both the Tropicks: and so may you make any other of the parallels. I do not advise you to draw the Meridians and Parallels in this form, least you cumber your Instrument, but I shew you how you may represent any of them in a moment, when ocasion requireth.

The Meridians of the Mater, (that were so called in the Meridional Projection) are here turned into the severall Hori­zons of the World. And the Parallels here serve only to gradu­ate those Horizons. Out of these Horizons choose your own Horizon, and distinguish him if you will that you may readily find him when you shall looke for him.

Your Horizon is thus inquired. Because the Elevation of the Pole at Northampton is 52. degrees 15. minutes, therefore [Page 36]from the Center (now North Pole) number in the Meridian line Northward 52. degrees 15. minutes, and there cutteth the the North Semicircle of our Horizon, or there you may Imagine him between the 52 and 53 Horizons, and the Southern Semi­circle thereof lies 52 degrees 15 minutes on the other side the Center towards Meridies. This may seeme strange that the North and South points of the Horizon, which in the Sphear are unequally distant from the North Pole, viz. the one but 51. degrees 15. minutes, and the other 127. degrees 45. minutes, (the supplement thereof) should be equally distant in this Pro­jection. But the reason is because the Center is both North and South Pole here at pleasure, and the Northern and Southern Hemisphears are both here represented by turns. Carry this in your head, and then lay the Eabel upon the South part of the Meridian, and number thereon from the Center (now North Pole) outward to the Equator at the Limb 90. degrees, thence number backward toward the Center (now the South Pole) the Elevation of the Equator (which is always complement of the Elevation of the Pole, and is here 37. degrees 45 minutes) there is the Southern point of the Horizon, and is distant from the Center (now South Pole) onely 52 degrees 15 minutes, but from the Center being North Pole 127. degrees 45. minutes, and from the Northern point of the Horizon before found just 180. degrees, as it is in the Sphear. Having found the North arch of your Horizon 52. degr. 15. min. behind the Center; count as many degrees and minutes forward in the Meridian before the Center toward Meridies, and the arch crossing there shall be his match to make up the whole Circle; and so may you find your whole Horizon upon the Mater whatsoever your Latitude be.

Here you must remember, that Stars which have Northern Declination rise and set upon the Northern arch of the Horizon; and those which have Southern declination upon the Southern arch. Remember also, that many Stars between the Tropicks which have Northern Latitude, have nevertheless Southern De­clination, and contrary many which have Southern Latitude have Northern Declination.

The lineaments of the Reet serving you in this Projection, are onely the Ecliptick, and the fixed Stars, the Almicanters and Azimuths here are of no use. The Meridians and Parallels [Page 37]are supplyed by the Label, for the Reet as well as for the Mater.

And whereas the Ecliptick here seemes to be irregular, see­ing the Solstitial points of Cancer and Caprcorn are not distant 180 degrees, as they should be, you must imagine that the South­ern arch of the Ecliptick is Projected by the eye placed in the North Pole, and for the Northern arch the eyes place in the South Pole: and the Center serveth for both the Poles alike, as hath been shewed: number therefore as you were taught for the Horizon in this Projection. For the reason of the draught of the Horizon and of the Ecliptick in this Projection is the same.

CHAP. III. Of the Nonagesimal Projection, shewing the Eastern and Western parts of the Sphear, being divided by the Azimuth of the Nonagesimus gradus.

NUmber in the Limb from the Equinoctial line toward the Pole the Altitude of the Nonagesimus gradus, (which is the highest degree of the Ecliptick) and thereto set the Finitor, turning the Almicanters either to the North or to the South, as your work proposed shall require.

Now is the Finiter Ecliptick, his point at the Limb-is Nona­gesimus gradus. The Center of the Planisphear is Ascendant and Descendant, the East and west points of the Horizon are here distant from the Center as much as the Amplitude of the Ascendant cometh to, to be counted from the Center upon the Eqinoctial line of the Mater, which here stands for Horizon: the Meridians and Parallels of the Mater are here Azimuths and Almicanters, but the Azimuths must be numbred from the East point of the Horizon. The Azimuths and Almicantars of the Reet, are here Circles of Longitude and Parallels of Latitude. Here are no Meridians nor Parallels of declination in this Pro­jection, onely the great Meridian of your place is to be found here, because he is an Azimuth as well as a Meridian, for he passeth through the Zenith as well as through the Poles of the World, and this Meridian is always distant form the Limb (or Azimuth of the Nonagesimus gradus) as much as the Ascen­dant [Page 38]is distant from the East point of the Horizon: for the Ampli­tude of the Ascendant and the Azimuth of the Nonagesimus gradus are always equall, and as the Meridian cutteth the Hori­zon 90. deg. Westward, from the East point, so doth the Azimuth of the Nonagesimus gradus always cut the Horizon 90. degrees Westward from the Ascendant.

This Projection is of excellent use, for getting the Altitude and Azimuth of any or all the degrees of the Ecliptick at once, also for getting the Longitude and Latitude of Planets, Comets, or Stars unknown, by their Altitude and Azimuth Observed.

CHAP. IIII. Of the Horizontal Projection, representing the upper and lower Hemisphears.

HEre the Limb must be reputed Horizon, and the Center of the Planisphear the Zenith of your place. Then may you by one of the Azimuths of the Reet represent the Ecliptick in any of his postures, (whatsoever degree be Ascending) and by the Label you may presently find the Al­titude and Azimuth of every degree thereof. Likewise may you here represent the plain of any Declining-inclining Dial, by some one of the Azimuths: and the Meridian of the Plain by one of the Meridians: by the help whereof you may resolve di­vers Problemes in Dialling, as shall appear in due place.

CHAP, I, Of the kinds and parts of Spherical Triangles.

IT is to be known, that 1, Spheri. Triang. are

• Rectangular
• Obliquangular.

2 A Rectangular Triangle is that which hath one or more Right Angles.

3 A Triangle that hath three right Angles hath alwaies his three sides Quadrants.

4 A Triangle that hath two right angles hath the sides oppo­site to those angles Quadrants, and the third side is the measure of the third angle. So that of those Triangles which have more Right angles, seldome ariseth any question. But the Right angled Triangle with one Right and two Acute angles, is that which comes most commonly to be resolved.

5 A Right angle is that which containeth 90 degrees, or openeth to one quarter of the circumference of any Circle de­scribed from the angular point.

6 All Spherical Triangles not Rectangled are called Oblique­angled. And if they have one angle greater then a Right angle, they be called Obtuse-angled, otherwise they be Acute-angled.

7 In rectangled Triangles the sides including the Right angle be called Legs, the side subtending it is called Subtense or Hypo­tenusa.

8 Either Leg of a rectangled Triangle may be made Basis, (if you will imagine him to lie level) and then the other leg shall be called Cathetus or Perpendicular.

[Page 40]9 In Oblique angled Triangles, the sides comprehending the angle given or sought, are called Legs, and the third side the Base.

10. In every Spherical Triangle there be (beside the Area or space contained) six containing parts viz. three sides, and three angles: of those six there must be three alwaies known or given to find out the rest.

CHAP. II. Of the 16 Cases of Rectangled Triangles. And how they may be reduced to five Problemes.

IN the Rectangular Spherical Triangle there be five parts one­ly come into the Question; the three sides, and two acute angles: because the third (being right) is alwaies known. Of these five parts any two being given the rest may be found.

There be 16 Cases or Problemes about Spherical Triangles. six for finding the Legs: four for the Hypotenusa: and six for the Angles. See Gellibrand. Trigonom. Britan. But we may here reduce them all to 5 Problemes, because our Planisphear re­solveth alwaies two of them at once. For by two parts given you shall presently get two of the three that are unknown: and if you do but turn the Triangle, you may presently have the third also: as shall appear in this book.

The Rectangular Triangle in question shall be marked throughout this book with A B C in this maner: so that the Base shall be marked and called B A, the Ca­thetus, or Perpendicular Leg C A, and the Hypotenusa. B C.

The right angle A. the angle at the Base B. the angle at the Cathetus C. And note that in the 4 first Problemes B A shall be alwaies set upon the Equinoctial line of the Mater, C A in a Meridian, B C on the Label, and B alwaies at the Center; as you shall find in the four next Chapters.

CHAP. III. PROBL. I. The Legs given, to find the rest.

SEt B at the Center, B A, on the Equinoctial line of the Mater, C A, on that Meridian which crosseth at A; then lay the Label to C, and your Triangle is made. Example. Let B A be 57. degr. 48.

min. C A 20 degr. 12. min. I number in the Equinocti­al line from the Center to­ward Meridies the length of B A to 57. degrees 48. minutes, and there I ima­gine stands A. Now be­cause this Base endeth be­tween the 57 and 58 Me­ridians, but neerer to the 58, I must imagine a Me­ridian passing between them in his due distance which shall cut off your Base in his due length of 57 degrees ⅘, and in that imagined Meridian I number upwards from the E­quator, by help of the Parallels, the length of the Cathetus C A 20. degr. 12. minutes; at the top of this Cathetus is the place of C, whereto I set the Label, and the Triangle is made upon the Planisphear, in such form as the figure here sheweth.

Now may you number B C the Hypotenusa upon the Label, and find it 60. degrees, the angle B hath his measure on the Limb between the Equator, and the Label 23 degrees ½.

Lastly to find the angle C do but turn the Triangle, setting C A on the Equinoctial, and B A on a Meridian, (according to the 1.8.) and laying the Label to B you shall find B C 60, degrees as before, and the measure of the angle C between the Equator and the Label may be reckoned on the Limb 77. degr. 43. min.

Note here that in this and all other like cases, in stead of the Label you may better use one of the Semidiameters of the Reet; for they have the same graduation, and lie closer to the Mater.

CHAP. IIII. PROBL. II. A Leg and the Hypotenusa given to find the rest.

SEt the Hypotenusa B C on the Label from the Center: the given Leg (to be marked C A) in one of the Meridians. Example. In the former Triangle, where the Hypotenusa was 60. degrees, the Cathetus 20. degrees 12. minutes. I number on the Label from the Center 60 degrees, and there make a prick for C; then I turn this prick to the Parallel of 20. degrees 12. minutes, and the Triangle is made. For the Meridian cutting this prick is Cathetus, and hath betwixt this prick and the Equator 20. degrees 12. minutes, as the Pa­rallels shew. Where this Cathetus cuts the Equator stands the right angle A, and between A and the Center lies B A, 57. degrees 48. minutes: the measure of B is on the limb between Meridies and the Label 23 degrees 30. minutes.

And for the angle C turn the Triangle. Set C now at the Center, calling it B, and you may find this angle as you did his fellow.

CHAP. V. PROBL. III. The Hypotenusa and an Angle given, to find the rest.

SEt the Hypotenusa on the Label, the Angle given at the Center. Example. In the former Triangle the Hypotenusa was 60. degrees, and the greater angle 77.43. I number in the Limb from Meridies toward Oriens 77. degrees 43. minutes, and there I set the Label, then I look the 60. degree of the Label numbred from the Center, at that 60 degree is C where the Hypotenusa and Cathetus meet. Here therefore the Meridian that cuts the Label in 60 degrees makes the Ca­thetus: I follow him down to the Equator and find his length 57 degrees 48 minutes, and from the point where he cuts the Equator I go straight to the Center, and find 20 degrees 12 mi­nutes the length of B A.

Lastly for the angle C turn the Triangle, setting C at the Center, and calling it B; and you shall find C as chap. 3. For [Page 43]whereas your Hypotenusa is 60, and your Cathetus 20 degrees 12 minutes, lay the 60 degree of the Label upon the 20 Parallel, and the Label shall cut in the Limb 23 degrees 30 minutes, the measure of the angle C.

CHAP. VI. PROBL. IIII. A Leg and an Angle given to find the rest.

IF the Leg be conterminate or adjoyning to the angle given, then make the given Leg Base, setting it upon the Equator; and move the Label from the Equator toward the Pole, so many degr, as the given angle B comes to. Then mark what Meri­dian cuts the end of the Base, that Meridian makes the Cathetus: follow him till he crosseth the Label, in that crossing is the angle C of your Triangle, from whence you reckon the length of the Cathetus to the Equinoctial, and the length of the Hypo­tenusa next way from C to the Center.

And now having all the sides, to get the angle C you shall turn the Triangle, and get him as in the 3 Chapter.

But if the Leg be opposite to the Angle given, make the given Leg Cathetus, that the angle given may be at the Center.

Example. In the former Triangle I have given the less an­gle 23 degrees ½, and the Leg opposite thereto 20. degr. 12. min. I open therefore the Label from the Equator to 23 deg. 30. min. on the Lamb, and mark where he cutteth the 20 ⅕ Parallel, for there is the angle C of the Triangle; thence you shall have a Me­ridian for the Cathetus going to the Equator, whose length is 20 degrees 12, minutes: thence in the Equator to the Center is the Base 57 degrees 47 minutes; and thence in the Label to C again, is the Hypotenusa 60 degrees.

And now having all the sides, to get the angle C, you shall turn the Triangle, and get him as in the 3 Chapter.

CHAP. VII. PROBL. V. The Angles given to find the Sides.

THis should have been the third Probleme, considering what is given. But because the way of resolving this [Page 44]case differeth from all the former, therefore I have reserved it to the last place.

In plain Triangles this case is insoluble, But in Spherical Triangles it may be resolved on the Planisphear, two wayes.

1. The first way hath M. Elagrave 5, 24. He sets B C on the Limb between the Pole of the Mater and the Vertex of the Reet, B A and C A one on a Meridian, and the other on an Azimuth crossing one another at a Right angle within the Limb. Which to do you must work thus. Example. In the former Triangle, the Angles are given A 90, B 23, degr. 30. min. C 77, degr. 43. min. Now first I will guess that the Hypotenusa is 40 degrees, and setting the Zenith 40 degrees from the Pole, that arch of the Limb between Pole and Zenith I take for my Hypotenusa, yet unknown. At the Pole shall be B, and at the Zenith C. Then because B is 23 ½ I take the twentie-third Meridian from my Hypotenusa which is on the Limb, and be­tween that and the 24th I imagin a Meridian which shall make B A of my Triangle. Also because the Angle at C is 77, deg. 43. min. I take an imaginary Azimuth near the 78th, numbred from my Hypotenusa which is on the Limb, and that Azimuth shall make C A of my Triangle. Now have I the Angles B and C, and three arches of which all the sides of my Triangle shall be made: but whether B A and C A cross at Right Angles I know not, and therefore I know not yet certainly the length of any side. Now to make the Angle at A a Right Angle, mark where your Azimuth (which you have taken for C A) cuts the Finiter, and from that point number in the Finiter toward the Center to 90 degrees, and there is the Pole of your Azimuth; ( viz. 12 degrees 17 minutes from the Limb) make a prick with ink at that Pole, and then look whether your Meridian 23 ½ (which you took for B A) cut this Pole, which yet he doth not as you will find. Therefore turn the Reet till the said Meridi­an do cut this Pole of the said Azimuth, and then you may be sure that Meridian and Azimuth, (wherever they cross,) do make Right Angles. (By Pitisc Trigonom. 1, 57) Therefore now have you all the Angles set on the Planisphear, and thereby all the sides found, viz. B C in the Limb 60, degr. B A in the Meridian 57 degrees 48 minutes C A in the Azimuth 20 degrees 12 mi­nutes, as they ought to be.

2. The second way M. Oughtred useth: It is this. For the [Page 45]rectangled Triangle whose three Angles be given, you shall frame another oblique Triangle, whose sides shall be equal to the Angles of the first Triangle. And so the Angles of the second Triangle to be found on the Planisphear shall be equal to the sides of the first Triangle, which are inquired. The Triangle which serveth for an Example throughout this Book hath his Angles A 90 degrees C 77 degrees 43 minutes, B 23 degrees 30 minutes. Here to find the sides, for the Angle A, I take half the Axis of the Mater from the Center to the Pole, and that shall be a side of 90 degrees. For the lesser Angle B, I reckon upon the Label from the Center a side of 23 degrees 30 minutes, and at the end thereof make a prick, on the Label. I turn this prick upon the Parallel from the Pole 77 degrees 43 minutes, and the Meridian there Crossing shall be the third side of this Tri­angle. Follow this Meridian to the Equator, and from his cut­ting there to the Center is the measure of the lesser Angle of this second Triangle, which is equal to the lesser Leg of the first Tri­angle. 20 degr. 12 minutes: likewise the arch of the Limb from the Pole to the Label, is the measure of the middle Angle of the second Triangle, which is equal to the middle side, that is, to the greater Leg of the first Triangle 57 degrees 48 minutes. And now having found the Legs of the first Rectangled Triangle you may by the first Probleme find the Hypotenusa to be 60 degrees. The Rectangled Triangle used in the first way, and the oblique Quadrantal Triangle used in this second way, shall appear in such formes on the Planisphear as these figures following do express.

CHAP. VIII. How to represent and resolve the Cases of the four first Problemes of Spherical Triangles, divers other wayes.

ONe way hath been shewn for representing any Rectan­gled Spherical Triangle upon the Planisphear, by the Label, Equator-line, and a Meridian, and thereupon to find out the sides and Angles of any such Triangle.

Now for Variety sake, and for the exercise of Learners in the knowledge of the Sphear, and because the same Angle somtimes may be more distinctly represented in one part of the Planisphear then in another, I have thought good to set down six other wayes, by which the four first Cases of Spherical Rectangled Triangles may be pictured on the Planisphear, and resolved.

There be three places in the Planisphear where the Angle B may be placed, whether he be given, or sought.

• 1. At the Center, and there his quantity is measured by the Label or any Semidiameter of the Reet, moving upon the Ring. thus was B placed in the former Chapters, and shall be once more in the first Variety.
• 2. At either of the Poles of the Mater, where by the Meri­dians that issue thence you may number the quantity of any An­gle from 0, to 180.
• 3. At the Zenith or Vertex of the Reet, where the quantity of the Angle may be numbred by the Azimuths in like manner.

CHAP. IX. The first Variety.

HEre the angle B shall be at the Center as before; B A on the Finiter, C A in an Azimuth, B C in the Axis of the Mater. So shall you have your Triangle pictured in the same form and quantity that he had in the for­mer chapters though other lines be here used. And to resolve the four first Cases of Rectangled Spherical Triangles with these Circles, you shall,

• 1. In the first Case where B A and C A are given, Number B [Page 47]A from the Center upon the Finiter; where it ends, you shall meet an Azimuth upon, which you shall number C A toward the Zenith; in the top of C A make a prick with ink for C, and then turn that prick to touch the Axis of the Mater. Thus have you all the sides in view, and the measure of the ngle B you shall find upon the Limb, between the Pole and the Fini­ter. And for C you must turn the Triangle as before hath been taught.
• 2. In the second Case. B C and C A given, Number B C in the Axis from the Center, and at his end for C make a prick; Then for C A count to what Almicantar he will rise from the Finiter, and turn the Reet till that Almicantar cut the prick C in the Axis; and the Azimuth there crossing the Axis and Al­micantar in C shall make C A. And between that Azimuth and the Center shall be B A on the Finiter. B shall be measured as before. C shall be found as before.
• 3. In the third Case B C and B given. Set the Finiter as much from the Pole as the angle B comes to. Then number B C in the Axis from the Center B to C. Thence turn down in the next Azimuth to the Finiter and you make C A. Thence turn to the Center, and you close the Triangle with B A. C shall be found as before.
• 4. In the fourth Case, if B and B A be given. Set the Finiter to the angle B, as in the third case in this chap. and from the Cen­ter upon the Finiter number B A. from A go up an Azimuth to the Axis, where C shall stand. From thence go to the Center, and you have compassed your Triangle, and all is shown by the view, but the angle C, which may be found as before. But if B and C A be given, set the Finiter to the angle, as in the third Case of this chap. then count in what Almicantar C A will end, and follow this Almicantar to the Axis, where they meet is the point C. And the Azimuth that cutteth there shall cut the Fi­niter in the place of A.

CHAP. X. The second and third Varieties.

SEt B at one of the Poles of the Mater where the Meri­dians meet; B A on the Limb either; way B C in a Meri­dian C A on the Label. One Example of the third Case shall suffice. B and B C are given. B is 77 degrees 43 mi­nutes; Number therfore the Meridians from the Limb till you come past 77 and almost to 78, and there imagin a Meridian to be drawn for the Hypotenusa of your Triangle; That Meridian maketh an angle of 77 degrees 43 minutes with the Limb, as the Hypotenusa of your Triangle doth with the Base. And be­cause the Hypotenusa B C is 60. therfore the 60th Parallel rec­koned from the Pole shall determin his length, and cut him off in the point C. Prick the point C (that is, the crossing of the 77 43 Meridian with the 60th Parallel from the Pole) and to that prick lay the Label: so that part of the Label which lieth between the prick and the Limb shall be C A, and the arch of the Limb between the Pole and the Label shall be B A of the Triangle. So shall all be known but C, which also may be found if you turn the Triangle as before.

Or thirdly, using only the Reet and Label, Set B at the Ze­nith; B A on the Limb of the Reet; A C in the Label; B C in an Azimuth, and you shall make the same Triangle on the Reet, that you made last on the Mater.

CHAP. XI. The fourth Variety.

SEt B at the Zenith of the Reet. B A upon the Limb from the Zenith to one of the Poles of the Mater, C A in the Axtree-line of the Mater, B C in an Azimuth.

CHAP. XII. The fifth Variety.

SEt B at one of the Poles of the Mater, B A upon the Limb between the Pole and Zenith, C A in the Axis of the Reet, B C in a Meridian.

Note here, that if you set the Triangle upon the Plani­sphear either of those two last wayes, you shall find him to be set both wayes, and that you haue your Triangle twice found, or two Trangles each of them representing the Triangle in Question; one is toward the right hand and the other toward the left: And they are both comprehended between the Axletrees of the Mater and Reet; and the arch of the Limb which lie: between the two Axletrees is Base to them both.

CHAP. XIII. The sixth Variety.

SEt the Zenith line of the Reet in the Equinoctial line of the Mater. Then set B at the Zenth B A upon the E­qiunoctial line inwards from the Limb, C A in a Me­ridian, B C in an Azimuth.

Thus have you various wayes for describing and resolving any rectangular Spherical Triangle upon your Planisphear. If in trying one way you find the points of your Triangle too much shadowed with the Reet, or that the sides cross one another too obliquely, that you can hardly find the point of the angle, then may you trie another way, and you shall likely find that fault amended.

These three last Chapters you shall easily understand, if you understand the former Chapters of this Book. And therefore I thought it needless to use any further examplification.

CHAP. XIV. Of the Solution of Oblique angled Spherical Triangles: And generally of all Spherical Triangles.

THese six Chapters following might well have been pla­ced before the second Chapter. For howsoever they best serve Oblique angled Triangles, yet are the Rules general, and may serve very well for the solution of all Spherical Triangles whatsoever. But I like this order well e­nough, and I think the Reader will have no cause to dislike it.

There be twelve Cases of Oblique-angled Spherical Tri­angles. But for the Planisphear they are here reduced to six. And they be all (unless I may except the last) as easily resolved upon the Planisphear as the five Cases of Rectangled Triangles.

Here note that for Oblique-angled Triangles, in all the Cases following, one side shall evermore be set upon the Limb between one of the Poles of the Mater and the Zenith, of the Reet, and the other two sides shall be made, one by a Meridian, and the o­ther by an Azimuth; at the meeting whereof is the Angle C, which onely may remain unknown after any Question resolved, and may be presently found by turning the Triangle, as before it hapned in the Rectangled Triangles.

PROB. 1. Three Sides given, to find the Angles.

SEt one Side, (which you will) upon the Limb between the Pole and Zenith, count the second side from the Pole by the Parallels, and count the third side from the Zenith by the Almi­canters: and know that where the last Parallel cuts the last Almicanter, there is the point of the third Angle C: the Meridian that passeth from this point to the Pole is the side A C: the A­zimuth that passeth from the same point to the Zenith is the side B C: and the third side A B is on the Limb between the Pole and Zenith. Now may you count the Angles at the Pole and Zenith and B: and for the third Angle C, turn the Triangle laying one of the other sides in the Limb between the Pole and Zenith, and you shall find that Angle also, as you did his fel­lowes. Note that whereas I called the sides of Rectangled Trian­gles [Page 51]B A. C A, and B C, that is Basis, Cathetus, and Hypotenusa, I choose here in Oblique-angled Triangles, to transpose the letters of the two first sides for diftinction sake, calling them A B, and and A C, and the third side indifferently either B C or C B.

Example. Let 40 degrees 70 degrees and 46 degrees ½ be Sides of a Triangle, whose Angles are sought. Now because I would first get the Angles joyning to the side 40 degrees. I mark that side A B, and set A B upon the Limb, A at he Pole, and B at the Zenith; which I remove 40 degrees from the Pole, accord­ing to the length of the side A B. Then because A C is 70 deg. I hold one finger (or a pin) upon the 70 Parallel from the Pole, and because B C is 46 ½ I hold another finger on the 46 ½ Al­micanter counted from the Zenith, and look where this Almi­cantar crosseth the said 70 Parallel, there is C of my Triangle: The Meridian that comes from the Pole to C is the long side of my Triangle A C; I count then from the side A B on the Limb how many Meridians lie between A B and A C, and I find that A C is just the 45 Meridian, therefore I say the Angle A at the Pole is 45 degrees. The Azimuth that comes from the Zenth to C is here the middle side of my Triangle, being in length 46½ I count from the side A B of my Triangle on the Limb how many Azimuths there are to this and I find that this is the 114 Azimuth almost, therefore the Angle B at the Zenith is al­most 114 degrees (exactly 113.30. minutes.)

Now to find the Angle C, I turn the Triangle, and set B C, 46 degrees ½ on the Limb (changing the letters into A B) And where the 40 Parallel crosseth the 70 Almicanter, there I meet with the 39 Azimuth, which shews me that the third Angle formerly called C, and now since the Tri­angle turned marked B is 39 degrees (exactly 38 deg 51 minutes.)

CHAP. XV. PROB. 2. Two Sides and an Angle comprehended given, to find the rest.

SEt the Angle given at the Pole and set one of the given sides in the Limb between the Pole and Zenith, the other given side you shall reckon on that Meridian which is distant from the Limb as much as the given Angle cometh to, at the end thereof there shall meet you an Azimuth which shall make the third side of your Triangle.

Example. In the former Triangle having A B 40. A C 70 and the Angle comprehended at A 45 degrees, I set A B on the Limb from Pole to Zenith, then because the Angle A is 45 de­grees I take the 45 Meridian reckoned from A B, and thereof I take 70 degrees (counting from A the Pole) for my side A C: at C in the 70 degree of this Meridian there crosseth an Azi­muth which makes my third side; this Azimuth is the 113 ½ be­ing numbred from A B therefore the Angle at B is 113 ½, and I finde between B and C in this Azimuth 46 ½ for the length of the side B C: onely C is now unknown, which you may also find by turning the Triangle.

CHAP. XVI. PROB. 3. Two Sides and an Angle opposite to one of them given, to find the rest.

SEt the given Angle

at the Zenith B, the side subtend­ing it in a Meridi­an A C, the other given side on the Limb A B.

Example. I have given A B 37 degrees 45 min. A C 105 degrees 41 minutes: and B 167 de­grees 09 minutes: I set the Zenith B 37 degrees 45 minutes from the Pole A, then because B [Page 53]is 167 degrees 9 minutes, I count the Azimuths from the side A B to 107 degrees, and farther 9 minutes, and I know that the Azimuth there imagined to pass (set between 167, and 168) shall make the side of my Triangle B C; but yet the length of B C I know not; and I want still a Meridian for the side A C opposite to the angle given. Now because A C his length is given (105 degrees 41 minutes, though the Angle A be yet un­known) I take the Parallel 105 degrees 41 minutes, numbred from the Pole A, and where this Parallel crosseth the 167 Azi­muth, there I am sure must be the Angle C: and the Meridian passing from C to the Pole is the side A C 105 degrees 41 mi­nutes: this Meridian lieth between the 12 and 13 number from A B, and sheweth the Angle A to be 12 degrees 26 minutes: the side B C, I may count 68 degrees ½ by help of the Almi­cantars. Now have I three sides and two Angles, which are more then enough to find the Angle C, when the Triangle is turned.

Note that you may place the known Angle at the Pole as well as at the Zenith, and it may be needfull so to do when the Angle C of your Triangle would otherwise fall under the Limb of the Zodiaque.

Note also that the Angle C may sometime fall under the Fi­niter, where the Azimuths faile. As if you had set the Angle 167 degrees at the Pole, the opposite side 105 degrees 41 mi­nutes had been set in an Azimuth, and C had been beyond the Finiter: your remedie in this case is to set Nadir in the place of Zenith, so shall C fall among the Azimuths just as you would have him. Example. Set 37 degrees 45 minutes between the Pole and Nadir (a b) count the Angle given at the Nadir b 167 degrees 9 minutes and his supplement 12 degrees 51 mi­nutes, for A C count a C the supplement thereof, and you shall find b C 111 ½, whose supplement is B C 68 ½.

Note thirdly, that if the angle given in this chapter be a cute, then if you place the known Angle at the Zenith, the Parallel may cross the Azimuth twice; or if you place the known Angle at the Pole, the Almicanter taken to find out the opposite side, may cross the Meridian twice; and so it may be doubtfull in which intersection the Angle C shall be found: That you may discover, if you examine which agrees best with the other parts of the Triangle being turned; or if you reduce this Triangle to [Page 54]two Right-angled Triangles, by letting fall a Perpendicular. Of which see the last Chapter.

CHAP. XVII PROB. 4. Two Angles and the Side comprehended between them being given, to find the rest.

SEt the side given between the Pole and Zenith on the Limb then count one Angle among the Meridians, the other among the Azimuths, and where the Meridian and Azimuth bounding the said Angles meet, there is the point of the Angle C, and all is known but the Angle C, which you may find also, if you turn the Triangle.

Example. In the 14th Chap. The Angle A was 45 B 113 ½ the side A B comprehended 40. Having set the Zenith 40 from the Pole, I seek the 45th Meridian from A B, and the 113 ½ A­zimuth from A B, and where they cross is C. Now may I num­ber A C by the Parallels 70 and B C by the Almicantars 46 ½. C may now be found by any of the 3 former Problemes, if you turn the Triangle, and set C at the Pole, or at the Zenith.

CHAP. XVIII PROB. 5. Two Angles and a Side opposite to one of them given, to find the rest.

SEt the Angles given, as A, and B, at the Pole and Zenith; the known side, as B C in an Azimuth; Count among the Meridians the Angle opposite to the known side, and having found the Meridian that boundeth him, lay a finger or a bodkin point thereon; then count the other Angle a­mong the Azimuths, and when you come to the Azimuth that boundeth him, because that Azimuth maketh the known side of your Triangle, you shall number his length from the Zenith, and at the end therof make a prick, then turn about the Reet till this prick in the Azimuth touch the Meridian before found; and then is your Triangle formed on the Planisphear, and all is known: but the Angle C to be found as in the former Chapters.

Example. Let be given A 45 degrees B 113 ½ B C 46 ½. I [Page 55]count from A B to the 45th Meridian, upon which I lay my finger, that he get not away for he must make my side A C, then I look the 113 ½ Azimuth (from A B) to stand for the given side: and because his length given is 46 ½ therfore in this 113 ½ Azimuth at 46 ½ below the Zenith I make a prick: then I turn the Reet till this prick touch the 45th Meridian, there at that touch must C stand; thence to the Pole is the side A C 70, and on the Limb I have the side A B 40. C is to be had by turning the Triangle, as in every of the former Problemes.

CHAP. XIX PROB. 6. Three Angles given to find the Sides.

THis Case comes very seldome in use. Yet that our Method of Trigonometry by the Planisphear may be compleat, and that no Probleme that is soluble may be left here unresolved, I shall shew the solution of this Probleme also. M ^r Blagrave, it seemes, never attempted this, con­tenting himself that he had found the way to resolve this Pro­bleme in Rectingled Triangles, which also he had once given o­ver as impossible. Blagr. Book 5, 24.

For resolving this Probleme it is to be known that if you go to the Poles of the 3 great Circles wherof your Triangle is made, these Poles shall be the angular points of a second Triangle; and the two lesser sides of this second Triangle shall be equal to the two lesser Angles of your first Triangle; the greatest side of the second Triangle shall be the supplement of the greatest Angle of the first Triangle (that is, shall have as many degrees and minutes as the greatest Angle of the first Triangle wanted of 180 degr.) see Pi [...]scus Trigonometry Lib. 1. Prop. 61.

This second Triangle therfore (all whose sides are known from the Angles of the first) you shall resolve by the first Probleme of Oblique angled Spherical Triangles. Chap. 14. And having by that Probleme found the Angles of this second Triangle, ‘know that the 2 lesser Angles of the second Triangle shall be severally and respectively equal to the two lesser sides of the first Triangle. (and the least Angle to the least side, the middle Angle to the middle side) and the greatest An­gle of this second Triangle being subtracted out of 180 degr. [Page 56]shall leave you the greatest side of your first Triangle.’

Example. If the Angles be given 113 ½ degr. 45 degr. and 38 degr. 51 minutes, and the sides be enquired. Draw by aime a rude Scheam of this first

Triangle, writing in the An­gle A 45 degr. in B 113 ½ in C 38 degr. 51 minutes, supposing these sides yet un­known: then draw un­der this by aime also a Scheam of the second Tri­angle,

setting his Base Pa­rallel with the Base of the first and making the Base of the second shorter then the Base of the first. Set also B at the Vertical Angle, and A C at the Base; as in the first Triangle. Then say,

Because A in the first Triangle is 45 degr. therefore in the se­cond Triangle B C (subtendeth A) shall be 45 degr. And because C in the first Triangle is 38 degr. 51 min. therefore in the second Triangle the side A B (which subtendeth C) shall be 38 degr. 51 min. And because B the greatest Angle in the first Triangle, is 113 ½ therefore in the second Triangle the side A C (which sub­tendeth B) shall be the supplement thereof, viz. 66 ½. Write now upon the sides of this second Triangle the quantities of the sides, so is your second Triangle ready to be resolved by the first Probleme of Oblique-angled Triangles whereby you shall find the Angles of the second Triangle, as I have expressed them in the Scheam. A 46, 26 min. C 40, B 110 degrees.

Now lastly I say these Angles of the second Triangle thus found, give me the sides of the first Triangle, which I seek, in this manner.

In the second Triangle. In the first Triangle.

• A is 46.26. Therefore B C is 46.26.
• C is 40.00. Therefore A B 40.00.
• B is 110.00 Therefore A C 70.00.

Supplement of 110 degrees. And thus by all the Angles given, we have found out all the sides, which was required.

Now would you see where this second Triangle dwells in the [Page 57] Planisphear, by whose help we have found out the sides of the first? That I will now shew you; because many may be as glad to know it as I was when I first found it. Having then the Angles of your first Triangle given, and his sides also now found; place him as in the 14 Chap. A B, 40 in the Limb. A C 70 in the 45th Meridian. B C 46 degr. 26 min. in the 113 ½ Azi­muth. Then

you shall say, Because the Center of the Pla­nisphear is the Pole of the arch A B, therefore at the Cen­ter shall stand the Angle C, which A B subtendeth: Next fol­low the 113 ½ Azi­muth (which maketh B C of your Triangle) to the Finiter, and from the point where he toucheth the Finiter you shall number in the Finiter to the Center 23 ½, and number on 66 ½ more beyond the Center to make up 90, and there is the Pole of the arch B C. Therefore there shall stand the Angle A, which B C subtendeth. Then follow the 45th Meridian to the Equator, and thence count in the Equator 45 degr. to the Center; and 45 degr. more beyond, which make 90: there is the Pole of the arch or side A C. There­fore there shall stand the Angle B which A C subtendeth. Here you see your second Triangle made by the Poles of the first ad­joining to the Center of the Planisphear under the Finiter: onely the side A B is wanting: To get that, prick A and B with ink on the Mater, if your Planisphear be metal; and when they be drie (if you can have patience to tarry so long.) turn about the Reet till some one Azimuth or other do cut both these pricks, [Page 58]which here the 48th or 49th from the Limb will make a shift to do (if your Zodiaque do not obscure one of the pricks) and in this Azimuth you may number between the pricks 38, 51, for the length of the side A B. Thus I have shewed you how all the sides and the Angle (C) of the second Triangle (made be­tween the Poles of the first Triangle) may be found in his pro­per place where he dwells in the Sphear, below the Finiter; and how to find both these and all the rest, by hoising up this Tri­angle to the Pole and Zenith, hath also been shewed in this Chapter.

CHAP. XX. How to reduce an Oblique angled Triangle to two Rect­angled Triangles, by letting fall a Perpendicular.

BEcause the third Probleme of Oblique angled Triangles cannot be resolved by the Canon of Sines and Tangents without letting fall a Perpendicular, and because in that case the crossing at the Angle C is oft so Oblique that you cannot define the Angular point certainly, and because in the Method for resolving the third Probleme one of the sides of the Triangle hapneth some times to make two intersections with the Parallels or Almicanters, and there may be doubt which of these intersections is to be taken for the Angle C. Although I there shewed another way to resolve that doubt, yet I will shew you also how to resolve it, and to remedie the inconvenien­ces aforesaid, by letting fall a Perpendicular. And it shall suffice to shew you this in one example, which if you mark and be acquainted with the four first Problemes of Rectangled Spheri­cal Triangles you shall be able to do it in any other needfull case whatsoever. Take therefore the Triangle of Chap 16. where we had given A B 37.45 minutes A C 105, 41. minutes. B 167. 9 min. you must observe that the Perpendicular ought to fall from the end of a known side, and to subtand some known An­gle, which here cannot be, because both the Angles at the Base A C are unknown.

Continue therefore the sides A C and B C to Semicircles, and you shall have a second Triangle N P C, in which, N P is equal to A B, N C is supplement of B C. P C suplement of A C: N supplement of B. C is common to both Triangles.

[Page 59]Here then in the

second Triangle N P C, let the Perpendicular fall from P upon the Base N C, so have you two Rect­angled Triangles, P R N, and P R C.

In the Triangle P R N you have (beside the Right Angle R) the An­gle N 12. 51 minutes, (supplement of B) and the Hypotenusa N P 37. 45 minutes; and so may find all the rest of this Triangle, by the third Probleme of Rect­angular Spharical Triangles, viz. P R 7. 49 min. ½ N P R 79, 49 min. and R N 37 degrees, which had,

In the Triangle P R C, by the second Probleme of Rectan­gular Spharical Triangles you may find R C 70 degrees (which added to R N maketh N C 111 ½, whose supplement is C B 68 ½) C 8 degr. ½, C P R 88 degr. which added to N P R maketh the whole Angle C P N 167, 47. which being subducted out of 180 degr. leaveth the supplement thereof C A B 12.13 min. as I find it by my Planisphear; and by exact calculation it may be 12 26 minutes.

Thus have you a perfect Method of resolving all Spherical Triangles by the Planisphear.

CHAP. I. The Preface.

THe best method (in my judgement) for setting down the Problemes of the Sphear is, to set them in such order, that the former may be Praecognita to the latter, and the latter pre­suppose the knowledge of the former. This most Authors have used. But this method here aimed at, perhaps is not alwaies kept exactly. Because where one Triangle serves to resolve di­vers Problemes, I was willing to make an end with him some­time before I meddled with another, for avoiding the multi­plicity of Chapters, and repetition of the same Schemes.

THere be in the Sphearfive famous Triangles, by the knowledge where of most Astronomical Problemes are resolved, insomuch that if you be but well versed in the general Problemes of Trygonometry set down in the former Book, and have acquaintance with these five Triangles in the Sphear, you will be able to resolve most of the following Problemes without any further help.

[Page 61]Of those five Triangles three are Rectangled, which shall be here denominated from their Hypotenusa's.

1. The Ecliptical Triangle, whose Hypotenusa is an arch of the Ecliptick, his Legs are arches of the Equator, and a Meridian: he serveth especially for Questions of the Suns Longitude, Right Ascension, and Declination, with some others, See this Ch. 6 &c.

2. The Horizontal Triangle, whose sides, are arches of the Horizon, Equator, and a Meridian. He serveth especially for Questions of the Suns Amplitude, Ascensional difference, and Declination, and of the Latitude of your place. See this Chapter 14.

3. The Azimuthal or Parrallactical Triangle, whose sides are arches of an Azimuth, the Ecliptick, and a Circle of Lon­gitude: he serveth especially to find the Moons Parallaxes in Altitude, Longitude, and Latitude. See this Chap. 64.

4. The other two are Oblique angled. One I use to call the Complemental Triangle, because all his sides be complements, viz. the Complement of Latitude, of Declination, and of Alti­tude. He serveth. cheifly to find the Altitude, Azimuth, and Hour. See this Chap. 24.

5. The last, I use to call the Polar Triangle, because one side of him is evermore the distance of the Poles of the World, and of the Ecliptick (23 degrees ½) his other sides are a Meridian, and a Circle of Longitude. He serveth chiefly to find the Lon­gitude and Latitude, the Right Ascension and Declination of the Stars. See this Chap. 34.

CHAP. II. How to find the Altitude of the Sun or Stars, by Obser­vation, with the Planisphear. Also what fashion is best for Sighst.

THe Planisphear may here supply the office of a Qua­drant (which is the fittest and most common Instrument for taking Altitudes) For the Planisphear is divided in­to 4 Quadrants, and if you hang a plumb-line at the Center, it may serve any of them. Set your Sights to one of the Semidiameters of the Mater, and turn him so to the Sun that the Sun may shine through the Sights; then shall the plumb-line (if [Page 62]it hang Parrallel to the Planisphear, neither bearing upon it, nor hanging off from it) shew in the lowest Quadrant of the Limb the degrees of Altitude. But because the Quadrants may be smal, I have shewed you a way how to make them serve your turn as well as if they were of double Semidiameter, Book 1.10. whither I refer you.

My Sights for the Sun and Moon I have devised to make thus.

Let them be about an inch square for a Planisphear of a foot Diameter: And in the middle of that sight next you (which must be a thin plate) let a very smal hole be drilled quite through: in the middle of the Sight next the Sun, bore an hole as big as a Pease, or bigger, whose Center must answer the smal hole in the other Sight, then cross the Center of the hole in the Sight next the Sun with an hair, or fine thrid, so that the thrid may run level or Parrallel with the Horizon, when you use the Sights. When you turn the Sights toward the Sun, and the shadow of the thrid fall, upon the smal hole of the lower Sight, you shall set or hold a white paper about a span behind the lower Sight, upon which paper you shall perceive a smal Image of the Suns body, and likewise of the thrid cutting through the midst of him very distinctly. And here you shall observe that the image of the thrid moveth upon the image of the Sun in the paper, contrary to the motion of the shadow of the thrid upon the lower Sight; for when the shadow of the thrid toucheth the bottom of the Sight­hole, the image of the thrid shall touch the top of the Suns image on the paper, and contrarily. But when the shadow of the thrid cutteth the middle of the Sight-hole, then shall the image of the thrid always cut the middle of the image of the Sun upon the paper exactly and clearly. Also you shall observe that though the Diameter of the Sun be always more then 30 min. yet the Diameter of the image cannot be observed here to be much above 20 min. as you may measure by the min. of the Qua­drant which the plumb-line passeth over, while the image of the thrid passeth over the image of the Sun: whither the Diameter of the Suns image on the paper be diminished by reason of the thickness of the plate through which the beams pass, or because the image on the paper is smal, the beginning and end of the Obscuration by the image of the thrid, cannot be precisely ob­served, for the present, I leave to Optical men to enquire: Also what is the reason why the image of the thrid moveth contrary [Page 63]to the motion of his shadow, is a question of some difficulty: My resolution is, because that image is a species which passeth through the Sight-hole with the species of the Suns body: For when the shadow of the thrid falleth upon the lower part of the Sight-hole, then certainly the upper part of the Suns body is a­bove the obscuration of the thrid, and the lower edge is Eclipsed at the Sight-hole. Now the wayes of the Suns body thus Eclip­sed on the lower side, passing through the Sight-hole, must needs be there decussated, so that the wayes or beames comming from the lower part of the Sun shall make the higher part of the image on the paper, and contrarily; as appeareth when an Eclipse of the Sun is observed by a Telescope, or by a smal hole, letting the beames into a dark room: For the reason here and there is the same. I have used those Sights for the Sun and Moon almost these 20 years past, and (for ought I could ever read or hear) they are of my own invention, and I have not met with any device more commodious to me for this purpose.

For the Moon, you must set your eye to the lowest Sight-hole, and let the thrid cut the middle of her body. For the Stars, if your eye cannot discern them by the thrid, you must behold them by the edges of the Sights both above and below. Or if you would observe the Stars Altitude by some larger Instrument, I advise that the Sight next your eye be a broad plate 4 or 5 in­ches square, in the middle whereof you shall cut a window whose length may be near 2 inches, and his breadth or height about an inch or more, so that your eye may be well shadowed, and yet have free scope through the window to find the Star. Let the up­per Sight be a Cylinder or ruler set Parrallel to the lower Sight, and his breadth be equal to the window almost, but narrower by a few minutes as 2 min. or 4 min. when you looking through this window can see the Star appear alike on both sides, the up­per Sight, then is your Instrument right set, and the plumb-line shall sh [...]w you his Altitude as before. Note that for all curious observation of the Sun or Stars, your Instrument must be suppor­ted with a Tripos, or like device, that it may be steddy, and that the apparent Altitudes of the Sun and Moon must be corrected according to the Table of Parrallax and Refraction. The sixed Stars have Refraction, but no Parrallax sensible. The quantity of the Parrallax is to be added, and the quantity of the Refraction to be subtracted alwayes from the apparent Altitude found, so shall you have the true Altitude.

[Page 64]Here followeth an Abridgement of Lansbergius Tables of Refra­ction and Parrallax of the Sun, as much as this Instrument may need, for the rest go to Lansbergius or Tycho Brahe's Tables at large, where you shall find the Moons Parrallax in the Horizon, to be sometimes 51 minutes, sometimes 1 deg. 7 min. at 70 degrees of Al­titude, between 18 deg. 24 min. Here Refraction is as the Sun.

CHAP. III. To finde a Meridian line.

STrike a straight line upon a Table or any Horizontal plain: and lay your Planisphear so that one of the Di­ameters of the Mater may lie in that line. Then take the Suns Altitude: the Altitude would be taken at least 2 hours (the more the better) before noon: and note, that if you take it between 29 and 30 degrees you shall be troubled nei­ther with Parrallax nor Refraction, because the Suns Refract [...]n and Parrallax be equall at the Altitude 29 degrees 26 minutes. The Altitude taken, you shall immediately lay your Planisphear in the posture aforesaid; and turning the Label to the Sun, make a prick in the Limb where the Label cutteth: And when the Sun comes to the same Altitude after-noon, your Planisphear laid as before, turn your Label to the Sun, and where he cuts make a second prick in the Limb. Then divide equally the Arch of the Limb comprehended between the pricks: and to the middle thereof lay the Label, and it shall point full North and South. Look then through your Sights; and if you see any Steeple, Pinacles. Chimney, Tree, or such mark, at a good distance in the line of Vision, you may note him for a South-mark, or for want thereof set up a smaller mark neerer hand. But note also that this may best be done when the Sun is in or neer the Summer Tropick, for neer the Equator he changeth his Declination so fast, that it may cause you an error of a few minutes, unless you make allowance for it.

[Page 65]Note, that all lines Parallel to your Meridians are Meridians.

2. Another way.

Having taken the Altitude of the Sun, or a Star, at a good distance from the Meridian; presently lay your Plani­sphear flat, and turn the Label to the Sun, or Star, as be­fore. Then by the Altitude taken, get the Azimuth; (by chap. 24 or 27 of this book.) Then remove your Labet (Eastward; if the Sun or Star were Westward from the Meridian; or West­ward if the Sun or Star be in the East Hemisphear;) so many de­grees as the Azimuth cometh to, and your Label shall be in the Meridian.

3. A third way.

When the great Wain is seen under Cynosura, (the Pole Star) observe with your eye the distance of the Thill-horse, called Alioth, from the next wheel of the Wain and setting that distance (by aime) in 5 parts, observe by a plumb­line when Alioth drawes neer to be in the same Perpendicular with the Pole Star. For when he wanteth but one of those 5 parts to come into the Perpendicular, then is the Pole-star in the Meridian over the Pole in our age: at other times of the night the Pole-star may be 4 degrees wide, and in one hour neer the Meridian he changeth his Azimuth above one degree.

4. A fourth way.

Because the distance of the Pole-star from the Pole is now 2 degrees 30 minutes, and the Pole is in the cir­cle or line which passeth from the Pole-star neer Alioth, as be­fore; you may by guess cut off from that line 2 degrees 30 min. and in that Section you have the Pole at any time. This way may be used abroad in the fields, where you cannot stand upon exactness; and herein you shall miss very little, if you accustome your self to observe the distances of the Stars about the Pole.

CHAP. IIII. To Observe the Azimuth of the Sun or Stars.

LAy your Planisphear upon an Horizontal plain or Level, and his Meridian on the Meridian line of your Place, found by the last Chapter. Then turn your Label that the Sun may cast the shadow of one Sight upon the other, or directly towards it, or till the shadow of a plumb line [Page 66]cut both the Sights alike, then doth the Label shew the Azimuth in the Limb. For the Stars, you must so direct the Sights by your eye, that their edges may touch the Visuall line that comes from the Star to your eye: and if your long Sight prove too short, turn him toward your eye, and inlightning the shorter Sight by a candle held behind you, mark where the edge of the long Sight cuts both the edge of the short Sight, and the Star; for there is your Label in the Azimuth of the Star, which you may count on the Limb.

Note that if you seek the Azimuth to get the hour, you shall find it most easily when the Sun or Stars are neer the Horizon: and then you shall not be troubled with their Refraction. But there is most use of observing Azimuths neer the Meridian, be­cause there the Azimuth changeth apace, the Altitude very slowly: Yet if you may choose, choose to take Altitudes rather then Azimuths (so you come not within 2 or 3 hours of the Meridian) because the Sights serve all Altitudes with like facility, and you may sooner have a true plumb line any where, then a true Horizontall plain, and a true Meridian line.

CAAP. V. To find the Suns Longitude.

THe Longitude of the Sun is the arch of his distance from ♈ 0 in the Ecliptick: or it is the angle made at the Pole of the Ecliptick comprehended between the circle of Longitude passing through ♈ 0, and another Circle of Longitude passing through the center of the Sun: for the said arch of the Ecliptick is always the proper measure of this Angle. And because the Suns center never hath Latitude, therefore for the Sun you shall enquire the arch; but contra­rily, for the Stars which have Latitude, you shall require the Angle: and they be both (as was said) of one measure.

The Suns Longitude (Arch or Angle) is presently found by the Ephemeris upon the Limb of your Planisphear, for if you lay the Label upon the day of the Moneth, it shall cut the degree of the Signe also in which the Sun is, and that is his Longitude: in doing whereof, you shall observe the cautions given Lib. 1.8. to which I refer you.

[Page 67]Note here, that the Longitude of a place in Geographie is the Angle at the Pole of the World, comprehended between the first Meridian (passing by the hither side of S. Michals Island, which is the neerest of the Azores) and the Meridian of the Place: and this Angle hath his measure in the Equator.

CHAP. VI. The Suns Longitude, Declination, Right Ascension, any one of them given, to find the rest in the first Projection.

WHat the Suns Longitude is, hath been shewed chap. 5. His Declination is his di [...]ance from the neerest point of the Equator; and therefore is alwaies mea­sured in an Arch of that Meridian which hapneth to pass through the center of the Sun, and always cuts the Equator at right Angles, as do all the Meridians.

The Right Ascension of the Sun is the angle at the Pole of the World comprehended between that Side of the Colurus Equi­noctiorum which cuts the intersection of the Ecliptick with the Equator in ♈ 0, and the arch of another Meridian which passeth through the center of the Sun. And note, that this angle may increase above 180 degrees, even to 360 degrees, though every angle, properly so called be less then 180 degrees, and never more then 90 degrees comes into the Triangle: for if you num­ber backwards or forwards from either of the Equinoctiall points, you shall have like arches of Right Ascension answering to like arhces of Longitude and Declination; so that having found the Right Ascension in any one Quadrant, or the comple­ment therof, you shall find the whole Right Ascension from ♈ 0 by adding one, two, or three whole Quadrants to the Right As­cension found, or to the complement therof, as by the view of your Planisphear you shall presently know how to do better then by more words. Otherwise thus. The Right Ascension of the Sun is an arch of the Equator comprehended between the Vernal Equinox and that point of the Equator which riseth with the Sun in a right Horizon. A right Horizon is where the Equator passeth through the Zenith, and maketh right angles with the Horizon, and consequently, where the Poles have no Elevation: For from that posture of the Sphear in which the [Page 68]Equator riseth upright, is the term of Right Ascension borrowed: I would, if I might, call it rather Equation; because it is numbred on the Equator, and serves for the Equation of naturall days, and may as easily be found in any Sphear as in a right Sphear, since the Horizon of a right Sphear limits the Right Ascension only be­cause that Horizon falls in with a Meridian, and the Meridians do limit it in all parts and postures of the Equator, without any respect to the Horizon at all. But the old term hath so long in­ured, that I beleeve it will not be changed without better Au­thority.

These definitions premised, you shall know that these three arches, viz. of Longitude in the Ecliptick, of Right Ascension in the Equator, and of Declination in a Meridian, do make up a notable Rectangled-Triangle in the Sphear, The Ecliptical Triangle. like unto that which was made the common Example, in all the five Problemes of Rectangled-Triangles. Book 3, 3. &c.

For there B C is the

Longitude (in ♊ 0) 60, degr. C A the Declina­tion 20 degr. 12 min. B A the Right Ascension 57 degr. 48 min. A is known a right angle, B is known, the angle of the Suns greatest Decli­nation, which for our age is 23 degr. 30 min. Now if but one of the Sides be given, you may find the other two by the Problemes of Rectangled Spherical Triangles.

But to see your Triangle, and resolve him in his proper lines, Go to the Mater of your Planisphear, and take him there in the first Projection. There number 60 the Suns Longitude in the Ecliptick line of the Mater from the Center outward. Where 60 endeth, there is C of your Triangle, and the Meridian that meets you there is C A the arch of Declination; follow him to the Equator, and you shall find by his graduation he is 20 degr. 12 min. Long. thence turn in the Equator to the Center, and you make B A the Right Ascension 57 degr. 48 min. so have you [Page 69]the true picture of your Triangle in his proper place. Observe your Triangle now, and you may see A is a right angle, for at such angle all the Meridians cut the Equator. B is 23 ½, for such an angle the Ecliptick dayly maketh with the Equator, as the arch in the Limb comprehended between them shewes. Now take for given any of the three Sides, and you have the rest. Take the Longitude for given (and be it 60 degr. as be­fore, or 70 degr. or what you will) and you may find the Decli­nation, and Right Ascension as before. Let the Right Ascension be given; then setting a needles point in the end thereof A, you may thence in a Meridian trace out the Declination C A to the Ecliptick, and the Longitude B C thence to the Center, every Side being divided into his whole parts or degrees. If the De­clination be given, say, Because the 20th Parrallel almost must cut off C A (the arch of Declination) in C, therefore I follow the Parallel 20 ⅕ to the place where he cutteth the Ecliptick and there comes the Meridian that serves my turn; and I may go down by him to the Equator, (as you would go down a ladder counting the rounds or degrees as you go) and so on, round my Triangle, and I need no more. For observe it when you will in the use of this Planisphear, if you can find the way to go round your Triangle, you have all the Sides measured to your hand, and evermore one Angle also, most commonly two, and the angle C onely left unknown.

But admit the Sun be in ♌ 0, then is his Longitude 120, degrees, and he is come back from the Solstice in your Planisphear as many degr. as he wanted of it before. Here the Triangle is equal to the former, and resolved in like manner. The Decli­nation is the same as before: But the arches of Longitude and Right Ascension in the Triangle are supplements of the true Lon­gitude and Right Ascension; shewing what the Sun wants of the Longitude and Right Ascension 180, in ♎ 0. wherefore subtract the Base of the Triangle 57 degr. 48 min. from a Semicircle, or 180 degr. and you shall leave 122 degr. 12 min. the Right As­cension of ♌ 0. or number in the Equator from the Center the way in which the Right Ascension hath increased, that is first to the Limb (which here is Colurus Solstitiorum) 90 degr. then back again to A the Right angle of your Triangle, and you have 32 degr. 12 min. to be added thereto. The Sum is 122 degr. 12 min. the Right Ascension, as before: If you observe this Example, [Page 70]you will easily perceive, that when the Sun is past ♎ 0. the Tri­angle will be on the other side the Center, and between ♎ and ♑ you must add to the Right Ascension and Longitude found within the Triangle 180 degr. and in the last Quadrant between ♑ and ♈ (where the Right Ascension again increaseth inwards) you must add 270 degr. to the complement of Right Ascension found in the Triangle, and take the sum, or else subduct the Right Ascension found in the Triangle from 360 degr. and take the residue for the Right Ascension.

CHAP. VII. To do the same in the second Projection, more easily.

IN the second Projection where the Center is the Pole of the World, and the Limb Equator, you shall find the E­cliptick, fairly drawn upon the Reet and distinguished into his quarters and degrees. Remember now from the for­mer chap. that the Ecliptick. Equator, and a Meridian, must make your Triangle; and know that the Label supplieth the place of the Meridians.

If the Longitude or Right Ascension be given, lay the Label on the degree given (in the Ecliptick for Longitude, or in the Limb of the Reet for Right Ascension) and your Triangle is made, and you may presently see your desire.

If the Declination be given, consider in what quarter of the Ecliptick the Sun is, then number the Declination given upon the Label inwards, and where the numbring ends make a prick on your Label, then move the Label into the quarter where the Sun is, and lay the prick on the Ecliptick there, and your Triangle is made, wherein you may see the Longitude and Right Ascension desired. This needeth no Example.

CHAP. VIII. To find the Angle at the Sun, made between the Ecliptick and Meridian.

THis is the angle C of the former Triangle, and is the one­ly part which cannot be found in the former posture of [Page 71]the Triangle, neither in chap. 6 nor 7, but is easily had by con­version of the Triangle, as you may remember out of the third Book.

Take the Triangle of chap. 6, and make the Cathetus Base, for this turn: and by the 1 or 2 Problemes of Rectangled Trian­gles, you may find this angle to be 77 degr. 43 min.

CHAP. IX. To find the said angle of the Ecliptick, with the Meridi­an, by the Longitude, Declination, or Right Ascension, divers other wayes.

IN the Meridional Projection do thus.

If you have the Longitude given, count the distance of the Sun in that Longitude from the next Equinoctial point, and count so many degrees in the Arctick Circle from the Limb inwards: to the end of this numbring, lay the Label, and between the Label and Equator you have upon the Limb the lesser angle made between the Ecliptick and Meridian; the great­er angle is the supplement thereof. Also between the Arctick Circle and the Limb you may find the Declination on the Label, which is more then was required.

If you have the Declination given, count it on the Label in­wards, and make a prick where the number ends, then turn this prick upon the Arctick Circle, and the Label sheweth the lesser angle in the Limb, as before.

Example. I would know what angle the Meridian that cut­teth the Sun in ♉ 9 degr. maketh with the Ecliptick. I number therefore in the Arctick Circle from the Limb inwards 39 deg. and to the 39th degr. I say the Label, and it sheweth in the Limb the angle sought 71 degr. 20 min. and in the Label the Declination of ♉ 9 degr. viz. 14. 32 minutes: this is a good way. But that the Label at this 39th degr. cutteth the Pole of the Ecliptick (as M ^r. Blagrave saith Book 3, 40.) is not true: either M ^r. Blagrave or the Printer here mistakes. For the Pole of the Ecliptick lies 14. 24 minutes nearer the Axletree, as you shall find in the next rule.

2. Another way. Mark what is the Right Ascension of the point proposed, being counted from the next Equinoctial point (as of [Page 72]♉ 9 degr. the Right Ascension is 36.36 min.) count so many de­grees in the Arctick circle from the Axeltree: at the end of this number is the Pole of the Ecliptick. Lay the Label to him, and you shall make a Quadrantal Triangle, whose Sides shall be equal to the Angles of the former Triangle, which was made of the Longitude, Declination, and Right Ascension, of the point pro­posed: for the Right Angle you have a Radius or Quadrant of the Axis: for the Angle of the greatest Declination between the Equator and Ecliptick 23 ½, you have the arch of a Meridian be­tween the Pole of the Equator and the Pole of the Ecliptick: for the angle sought, you have the arch of the Label, between the Pole of the Ecliptick and the Center 71.20 minutes; as be­fore: the least angle of this Quadrantal Triangle is at the Center, and you shall find his measure in the Limb 14.32 minutes: that is the measure of the least Side of the former Triangle, viz. the Declination of the point proposed.

Here you see, If the Declination had been given, you should have set it in the Limb, between the Pole and the Label, and so had you made the same Quadrantal Triangle, and might have found on the Label between the Arctick Circle and the Center the measure of the angle sought: and likewise in the Arctick Circle between the Label and the Axtree-line the Right Aseensi­on, though it be more then was required. The reason hereof you may learn from Book 3.7.

CHAP. X. To find the point of the Ecliptick in which the Longitude and Right Ascension have greatest difference.

Move the Label. on the Polar circle till you find the de­grees of the Label between the Polar circle and the Limb to be equal to the degr. of the Limb between the Label and the Pole, so have you a Rectangled aeqaicrurall Triangle made by the Limb, Label, and the Meri­dian 46 ¼; like to that in the second Variety, Book 3.10.

Here the angle B at the Pole between the 46 ¼ Meridian and the Limb, is equal to the Longitude of the point sought 46¼, and ei­ther Leg is equal to the Declination thereof 16 ¼: Therefore I con­clude, that when the Sun is 46 ¼ in Longitude. (that is in ♉ 16 ¼) [Page 73]then his Longitude hath furthest out run the Right Ascension. Subtract now the Right Ascension of ♉ 16 ¼, which is 43 ¾ out of the Longitude 46 ¼ there remains 2 deg ½: which being con­verted into Time, is 10 min. the greatest inequality of Ascension in a Right Sphear.

CHAP. II. To find the Latitude of your Place, or the Elevation of the Pole above your Horizon, by the Meridional Altitude, and Declination of the Sun. Meridional Projection.

GEographers call the distance of a place from the nearest point of the Equator upon Earth, the Latitude of that Place, as the Latitude of London is 51 deg. 32 min. from the Equator Northward: the Latitude of S ^t Tho­mas Island upon the coast of Africk is 0 deg. 0 min. because the middle of that Island lyeth under the Equator. And because the Latitude of your Place, and the Elevation of the Pole above your Horizon, are alwaies equal, therefore the Elevation of the Pole is oft called Latitude of the Place, or Latitude simply: and so for brevity sake we shall often call it. But when we speak of the Latitude of the Moon or Stars, you must understand Astronomers thereby mean their distance from the neerest point of the Ecliptick.

To find the Latitude of your Place, get the Suns Decli­nation, by the 6 or 7th. and his Meridian Altitude by the second of this Book: Then find the parallel of the Suns Declination, North or South as the Declination is, and where it toucheth the Limb (here Meridian) there is the point where you observed the Sun at Noon; set the South end of the Finiter so many degr. below this point as the Meridian Altitude had, then is your Finiter set to your Latitude, and you shall find the measure of it between the Equator and the Zenith, (which is properly the Latitude) and the same measure shall you find between the North point of the Finiter and the North Pole, where it is more properly called the Elevation of the Pole.

Example. June 20 1651. I observed the Meridian Altitude of the Sun, here at Ecton, four miles Eastward from Northampton, 60 degr. 59 min. the Longitude of the Sun was then ♋ 8 degr. 19 min. ½, his Declination 23 degr 14 min. Northward. There­fore [Page 74]having found in the Limb the point where the Parallel 23 degr. 14 min. toucheth above the Equator, I put the South end of the Finiter 60 degr. 59 min. below that point, toward the South Pole, which done, I see the North Pole Elevated above the Finiter 52 degr. 15 min. and the Zenith of my Horizon like­wise to be removed from the Equator Northward 52 degr. 15 min. which is the Latitude of Ecton.

Note that you may best observe the Latitude when the Sun is near the Summer Tropick; for then you shall not be troubled with Refraction; and then the Declination varyeth slowly; which varyeth almost one minute every hour near the Equi­noctial.

CHAP. XII. To do the same by the Meridian Altitudes of the Stars about the Poles.

MAny of the Stars near the Northern Pole may be seen with us twice in the Meridian in one Winters Night: that is, one while above the Pole, and 12 hours after again below the Pole. As for Example, the Pole­star, called Alrucabe, about December 18 will be in the Me­ridian above the Pole at 6 of the clock at Night, and at 6 next morning he will be in the Meridian below the Pole.

Observe both the Meridian Altitudes, and add them together, half that sum is the Elevation of the Pole. Example. I obser­ved at Ecton the greatest Altitude of the Pole-star to be 54 deg. 45 min. and his least Altitude 49 degr. 45 min. the sum is 104 deg. 30 min. the half 52 degr. 15 min. the Latitude of Ecton: and here I have gotten also the Pole-stars distance from the Pole, and consequently his Declination which is the complement thereof, for the Latitude being subducted from the greater Altitude leaves the Stars distance from the Pole 2 degr. 30 min. and conse­quently shewes his Declination to be 87 degr. 30 min. which is 39 min. more then Gemma Frisius observed it, Anno Dom. 1547. for in our age the Pole-star approcheth about 1 min. nearer the Pole in every 3 years.

Note that these Stars which are distant from the Pole less then the Latitude, and more then the complement thereof, have their [Page 75]less Meridian Altitude in the North part of the Meridian, and their greater Meridian Altitude in the Southern part of the Me­ridian beyond the Zenith. Wherefore for them you shall take the complement of their greater Altitude, and add it to the North Quadrant of the Meridian, and if to that sum you add the lesser Altitude, the half thereof shall be your Latitude. But the nearer any Star is to the Pole, the fitter for this purpose, and therefore none better then Alrucabe, who is the nearest of all.

CHAP. XIII. To find the Declination of the Sun or Stars, by their Me­ridian Altitude, and the Elevation of the Pole.

This is done by the first, made of the Meridional Pro­jection, where having set your Finiter to the Elevati­on of the Pole, or your Zenith to the Latitude, (for as hath been shewed Chap. 11. all comes to one, and in doing either, you do both) and having observed the Meridian Altitude of the Sun or Star, number the Altitude observed upon the Limb of the Reet on the South or North side of the Pole, according as the Star was observed to be, and there shall meet you on the Mater his Parallel of Declination.

Example. I observed the Suns Meridian Altitude at Ecton, 20 deg. I look therefore where the 20th Almicanter toucheth the Limb, (the Finiter first set to the Latitude) and there meets at the Limb the 17 ¾ Parallel below the Equator: where­fore I say, the Sun declineth 17 degr. 45 min. Southward. A­gain, I observed the Star Alhaiot in the North part of the Me­ridian 6 degr. 42 min. high, I go to that Almicanter in the North quarter of the Reet under the Pole, and there meeteth at the Limb the Parallel 45 min. ½ of North Declination.

CHAP. XIV. To find the Oblique Asoension and Descension, and the As­censional difference of the Sun or any Star: by his Declina­tion, and the Latitude of the Place, Two several wayes, in the Horizontal Triangle.

THe Oblique Ascension is the arch of the Equator which riseth with the Sun or any Star in an Oblique Sphear, that is, a Sphear wherein the Equator maketh an Ob­lique Angle with the Horizon. This arch beginneth alwayes from the Vernal Equinox, but we seek the latter term or end thereof. To find this by Calculation, we use to find first the Ascensional difference, that is the difference of the Right and Oblique Ascension, or the arch of the Equator comprehended between the latter termes of the arches of the Right Ascension and Oblique Ascension of the Star, this difference for North Stars, we subtract from the Right Ascension, and the remainder is the Oblique Ascension; but for South Stars we add it to the Right Ascension to make the Oblique Ascension: and for Ob­lique Descension or Setting, contrarily, we add the Ascensional difference for North Stars, and subtract it for South: you shall see all plain in the Meridional Projection of the Planisphear, and the first Mode thereof, where the Finiter is set to the Latitude.

Example. I would know the Oblique Ascension of the Sun in ♋ 0 and the Ascensional difference, The Declination of the Sun in ♋ 0 is 23 degr. 30 min. our Latitude 52 degr. 15 min. I go to the North Parallel 23 degr. ½, which is the Tropick of Cancer, on the Mater, and following him to the Finiter, there I turn in the Meridian which cutteth there, and go down to the Equator under the Horizon, and make a prick here; I say, is the Right Ascension of the Sun in Cancer 0, for the same Meridian cutteth both these, and therefore both these points would rise at once in a Right Sphear, where the Meridians by turns successive­ly, become Horizon: but counting how many degrees are be­tween this prick and the rising point of the Equator, I find 34 degr. 10 min. this is the arch of Ascensional difference, which being subtracted out of the Right Ascension of ♋ 0 (which by Chap. 6 is 90 degr.) there remaineth the Oblique Ascension 55 [Page 77]degr. 50 min. And the meaning is, that whereas the Sun being in ♋ 0 in a Right Sphear, riseth with the 90th degree of the E­quator, in our Latitude, he riseth with the 55 degr. 50 min. of the Equator: the difference of these Ascensions is 34 degr. 10 min. add this difference to the Right Ascension of ♋ 0, and it ma­keth 124 degr. 10 min. the Oblique Descension, for the 124th degree of the Equator setteth with ♋ 0, and the point of the Suns Right Ascension shall in North Signes Set before him as much as it Riseth after him, and in South Signes shall Set after him, as much as it Riseth before him. This you may see plainly by the view of this Projection; if you imagine it one while to be the Eastern Hemisphear, and another while the Western Hemi­sphear, at your pleasure.

Take in the Scheme of the Horizontal Triangle annexed, so many Circles of your Planisphear as you shall use for this pur­pose, and moreover see here how the Ecliptick should lie in your Planisphear when ♋ 0 is rising, which the Planisphear in this posture cannot express.

[Page 78]The same way serveth for the Stars, for the Stars Parallel of Declination followed to the Finiter, shall bring you to C of the Triangle, as the Suns did, and then you know what to do.

A second way and more easie and pleasant, is by the Equinoctial Projection. Place the Sun or Star upon the East part of your Horizon, (in the North-east quarter, if the Declination be North; but in the South-east quarter, if the Declination be South; as you had direction, Book 2, 2.) and the degrees of the Limb by which ♈ 0 is gone past Oriens, or the six a clock line of the Mater, are the degrees of Oblique Ascension, subduct this out of the Right Ascension, if the Star be North, or out of this sub­duct the Right Ascension, if the Star be South, and the remainder is the Ascensional difference. But this subduction is made to your hand in the Planisphear.

Take the former Example. The Latitude here is 52 degr. 15 min. the Suns Declination in ♋ 0, is 23 degr. 30 min. as be­fore. Now See in this second figure of the Horizontal Triangle A B C, how the Circles lie in the Planisphear, set ♋ 0 on the North­east part of the Horizon at C, and you have before your eyes.

CHAP. XV. The Ascensional difference, Declination, and Amplitude, of the Sun or a Star, and the Latitude of the Place, any two of them given, to find the rest.

T the Amplitude or Ortive Latitude is the arch of the Horizon between the rising-point of a Star and the full East point. This is the Hypotenusa of the Horizontal Triangle, expressed in both the Schemes of the former Chapt. Now I told you Book 3.2. that if any two parts of a Rectangled Triangle be given with the Right angle, the rest may be easily found; observe then your Triangle A B C in the first Scheme of the former Chapter, and likewise in the Meridional Projection of your Planisphere, you shall see the very same. For the Finitor being set to the Latitude, C shall be where the Tropick of Cancer cuts the Finiter: the arch of the Meridian between C and the Equator is C A and the Declination: thence in the Equator to the center is A B the Base, and the Ascensional Difference B C in the Horizon is the Amplitude; B is the com­plement of Latitude; A is 90 degr. C is unknown, and we need it not, else, if you have read the third Book, I hope you can find him.

• 1. Admit now that the Declination and Amplitude be given, put the term of the Amplitude (I mean the point where it ends, counting from the Center) upon the Parallel of the Declination, and your Triangle is formed, and thereby the Ascensional diffe­rence and the complement of Latitude are discovered.
• 2. Or if the Declination and Ascensional difference be given, number the Ascensional difference from the Center downwards in the Equator: Then go up in a Meridian as many degrees as the Declination comes to, and to the point where you end (which is C) set the Finiter, so he is placed to your Latitude, and the Amplitude also is shewn.
• 3. Or if the Declination and Latitude be given, the Finiter be­ing set to the Latitude, follow the Parallel of Declination to the [Page 80]Finiter there is C, thence go down by a Meridian to A in the E­quator, thence in the Equator to Bat the Center, thence turn by the Finiter to C, and you have compassed your Triangle, and therefore have all known but C.
• 4. If the Latitude and Ascensional difference be given, the Finiter being set to the Latitude, count from the Center in the Equator to the end of the Ascensional difference, there is A: Go up thence in a Meridian to the Finiter; there is C: Go thence in the Finiter to the Center: there is B.
• 5. If the Latitude and Amplitude be given, the Finiter being set to the Latitude, count from the Center (B) in the Finiter to the end of the Amplitude (where shall be C) go down thence in a Meridian to the Equator, (where is A) thence in the Equator return to the Center B.
• 6. If the Amplitude and Ascensional difference be given, prick the end of the Amplitude numbred in the Finiter from the Center, and prick the end of the Ascensional difference, numbred in the Equator from the Center: then turn about the Reet till some one of the Meridians cut both these pricks, and that shall make up the Triangle.

Note, that for South Stars, or the Sun in South Signes, this Triangle lies on the South-side the center, and above the Fini­ter; but for North Signes it lies North of the center, and be­low the Finiter.

CHAP. XVI. To do the same in the Equinoctial Projection.

HEre serves the second figure of the Horizontal Triangle in Chap. 14. where B A is the Ascensional difference; C A the Declination, B C the Amplitude, B comple­ment of the Latitude.

If the Latitude and Declination be given, number the Decli­nation on the Label inwards, and at the end make a prick, turn this prick to the Horizon of the Mater, and so shall the outward arch of the Label, be C A, the shorter arch of that Horizon B C, and an arch of the Limb B A of your Triangle.

If the Latitude and Amplitude be given, do as in this Exam­ple. I observed Sirius to rise 27 ¼ from the East South-ward [Page 81]my Latitude is 52 degr. ¼. I go to the 52 ¼ Meridian on the Mater, reckoned from the Center on the South-side, because the Star is Southern, as his rising shewes. This 52 ¼ Meridi­an being my Horizon (as Book 2.2.) I number in him the Am­plitude of Sirius, from Oriens toward Meridies 27 ¼, and thereto I lay the Label; and I see my Horizon cuts the Label in 16¼, that is C A the South Decimation of Sirius: and between the Label and Oriens in the Limb, I have B A 22 ¼, his Ascensional differ­ence. If you can do these two, you may resolve the four other Cases of this Chapter with like facility. View but the Scheam in the Book, and in your Planisphear, and that alone will instruct you.

CHAP. XVII. To find the Semi-diurnal and Semi-nocturnal Arches of the Sun or Stars: the time of their Rising and Set­ting: and the length of their Day and Night: by Declination, and the Latitude of the Place.

SEt the Finiter to the Latitude, (as-in the first Mode of the Meridional Projection.) Then seek the Parallel of the Declination of the Sun or Star, North, or South, as it hapneth to be. That Parallel shall be divided by the Finiter into two arches: the arch above the Finiter is the Semi-di­urnal arch, in which you may count the time of Rising and Set­ting, and the Length of the Day: that below is the Semi-noctur­nal arch, in which you may reckon the length of the Night; or if your Question be of a Star, the time he spends under the Ho­rizon.

Example. In the first Scheme of the 14th Chapter, D E is the Tropick of Cancer, that is the 23 ½ Parallel of North Declinati­on: C E is the Semi-diurnal arch: C D the Semi-nocturnal. And you shall find in the Meridional Projection of your Planisphear those arches are divided by the Meridians; and the arch C E containeth 124 degr. 10 min. which turned into houres and mi­nutes, (accounting every degree 4 minutes of Time, and every 15 degrees an houre,) is 8 houres 16 min. 40 sec. half the length of our longest Day, and the arch C D containeth 55. deg. 50 min. that is, three houres 43 min. 20 sec. half the length of our short­est [Page 82]Night: therefore at three hours 43 min. after midnight the Sun Riseth in the Tropick, and sets so much before midnight, that is, at eight hours 16 min. 40 sec. and so may you find your de­sire in any other Parallel.

Example. 2. I observe that Fomahant his Meridian Altitude is but 6.30 min. therefore by Chap. 13 he declineth South­ward 31¼. I would know how long he shines with us; and I presently see in the Meridional Projection of my Planisphear, that his Parallel hath but 38 degr. above the Horizon; that is, he will set two hours 32 min. after he is South; and the whole time he shines in our Horizon, is five hours four minutes.

Example. 3. Lyra her Declination is 38.30 min. North; and I see his Parallel comes within 45 min. of the Horizon, in the North part of the Meridian, but never toucheth it: therefore I conclude that Lyra never sets with us at all.

CHAP. XVIII. To find the same, in the Equinoctial Projection.

TUrn about the Reet till the Suns place in the Ecliptick, or the point of the Star, touch your proper Horizon: and that on the North side, if the Declination be North, or on the South side, if it be South. Lay the Label to the Sun or Star in the Horizon, and between the Label and Me­ridies upon the Limb you shall have the Semi-diurnal arch, both in degrees, and in hours and minutes. And you shall observe that those Stars whose Declination is greater then the comple­ment of your Latitude (as Lyra's was in the last Chap.) will ne­ver touch the Horizon at all. For Stars of such Declination, if they be North, never set; and if they be South, never rise at our Town.

But what shall I do if the Star be not in my Reet? Then will I number his Right Ascension on the Limb of the Reet, and having thereto laid the Label, I will number his Declination upon the Label from the Limb inwards, and where it ends make a prick by the edge of the Label, in the Reet, for him: for there is the place of the Sta: but if the Stars place happen to be in a window of the Reet, where the Reet is perforated, then I will make the prick upon the Labels edge at the Stars Declination, [Page 83]and turn that prick to the Horizon. I may pintch the Label close with the Reet, and turn both together, which willbe the handso­mer way, but if I move the Labels prick alone to the Horizon, it is sufficient for this Probleme, which needeth no more words.

CHAP. XIX. To find the beginning and end of Twilight, by the Suns Declination, and the Latitude of the Place.

SEt the Planisphear in the first Mode of the Meridional Projection; then turn the Planisphear that the Zenich may be downwards, and the Almicanters mostly be­low the Horizon. Then go to the 18th Almicanter be­low the Horizon: and wheresoever the Parallel of the Suns Declination doth cut that Almicanter, there is the beginning and end of Twilight: and because every Parallel is divided by the Meridians into 12 hours, or 180 degr. (every 15 degr. being one hour) therefore you may easily count how far the point where Twilight begins, is distant from Midnight, or from Noon, or from Sun-rise, or Sun-set in the Horizon.

Example. In our Latitude 52 degr. 15 min. my Planisphear set as aforesaid, I find that where the 18th Almicanter cutteth the Equator under the Horizon, there cutteth also in the same intersection the 30th Meridian, or second Hour Circle from the Axis and Center; by which I gather, that when the Sun is in the Equator, the twilight begins two hours before 6 or Sun-rising, and ends likewise at 8 of the clock at Night, the Sun then setting, (as you may see) at 6. Likewise where the Winter Tropick cuts the 18th Almicanter, there cuts also the first Meridian from the Axis South-ward; shewing that in the depth of Winter Twi­light begins 4 minutes after 6 in the morning, and lasteth till 5 hours 56 minutes after-noon. Likewise I see that about the be­ginning of ♊ where the Sun declineth North-wards about 20 degrees, the Twilight lasts till mid-night, and that from that time till the Sun comes to ♌ (that is, from May 11 to July 11, or thereabouts,) we have no dark night at all, unless the Skie be Cloudy, for in all that time the Sun is never found above 18 degrees under the Horizon.

CHAP. XX. To find the time of the Cosmical Rising and Setting of the Stars, by their Declination and Right As­cension, and the Latitude of the Place.

AStar is said to rise Cosmically when he riseth at the same instant with the Sun.

To find it, use the Equinoctial Projection: Turn the Star (being found in your Reet) to the East part of your Horizon, and look what degree of the Ecliptick cutteth the same East part of your Horizon; for when the Sun comes to that de­gree, the Star and Sun shall rise both together. If the Star be not in your Reet, put him in with ink, as you put in the rest, Book 1.7. if his place light upon a window, or hole of the Reet, prick him on the edge of the Label, and hold Reet and Label close together, while you turn him to the Horizon.

Example. Sirius I have among 40 other principal Stars in my Reet, and would advise you not to be without him, for he is a little Sun in a Winters Night, to tell you how the time passeth; he is called by the Latines both Canis, and Canicula; for they had no name for the little Dog, but called him by the Greek name Procyon, as Pliny witnesseth Lib. 18 Chap. 28: yet I have seen a late Writer, who takes upon him to teach the Col­ledge of Physitians both Physick and Astrologie, before he hath well learned either of them: who in his Obtrectations upon the Pharmacopaea Lond. in the Chapter of Vinum Scilliticum Gale­ni, betrayes his ignorance herein, as elswhere in 100 other things, which I could shew; for in that very Chapter pag. 147 of his fourth Impression, in 24 short lines he commits 5 absurd errors. 1. He makes it doubtfull whether Canis be to be taken for Sirius, or Procyon. 2. He goeth about to teach Galen where Squills grow; and that there is no hilly ground near the Sea. 3. He supposeth that the Acronycal rising of the Dog (which hapneth in the depth of Winter) is a fitter time to gather Squils, then the Heliacal rising, which hapneth near unto the Cosmical, in the heat of Summer. 4. He either supposeth that Squils grow in the Parallel of London, or that by the rising of the Dog at London men should gather Squils in Greece or Spain. [Page 85]5. He tells the Colledge that both the Dogs are between the E­quator and the South Pole, which indeed is newes, if it were true. Let the ingenuous Reader pardon this digression, and I proceed.

This Sirius I brought to our Horizon, (the 52 ¼) and found that there riseth with him in the Ecliptick ♌ 18 ½. in like man­ner, with Procyon riseth ♌ 6 ⅓: therefore Sirius riseth Cosmically with us August 1: and Procyon 12 dayes sooner. But in Greece and Spain, in the Latitude 38 degr. Sirius riseth with ♌ 4 ½, that is a fortnight sooner.

A Star is said to Set Cosmically, when the Sun riseth at his setting. Place the Star therefore on the West part of your Ho­rizon: then look what degree of the Ecliptique riseth in the East part; for when the Sun comes to that degree, the Star shall set Cosmically.

Example. I brought Sirius to the South-west part of our Horizon, where he useth to set. And in the South-east part I saw ♏ 23 degrees in the Ecliptique rising: therefore when the Sun is in ♏ 23. (which is about November, 5. then shall Sirius set Cosmically. But at Athens in Lat. 37 ¼,

• His Cosmicall Rising is ☉ in ♌ 4. July 17
• His Cosmicall Setting is ☉ in ♐ 9. Nov. 20
• The Pleiades in our Lat Cosmically Rise. ☉ in ♉ 12 ½ Apr. 22.
• The Pleiades in our Lat Cosmically Set ☉ in ♏ 29 ½ Nov. 11.
• At Athens Cosmically. Rise ☉ in ♉ 19 ½ April 30.
• At Athens Cosmically. Set ☉ in ♏ 27 ½ Novem. 9.
• Arcturus in our Lat. Cosmically. Ri. ☉ in ♎ 0 Sep. 13.
• Arcturus in our Lat. Cosmically. Set. ☉ in ♋ 4 June 15.
• At Athens Cosmically. Rise ☉ in ♎ 10 ½ Sept. 23.
• At Athens Cosmically. Set ☉ in ♊ 6. May 17.

CHAP. XXI. To find the time when any Star riseth or setteth Acro­nycally, by his Declination, and Right Ascension, and the Latitude of the Place.

WHen a Star Riseth just at Sun-setting, he is said to rise Acronically. To find the time, turn the Star to the East part of the Horizon in the Equinoctial Pro­jection, [Page 86]and mark what degree of the Ecliptique descendeth in the West: for when the Sun comes to that degree, the Star shall rise Acronically.

Example. When Sirius toucheth the South East Quarter of our Horizon, I see ♒ 18. setting. Therefore when the Sun is in ♒ 18. Sirius riseth Acronically.

A Star setteth Acronically, when he setteth with the Sun. To find the time, place the Star setting, in the West-part of the Ho­rizon, and see what degree of the Ecliptique setteth with him: for when the Sun is in that degree, the Star shall set Acronically. Thus in our Latitude.

• Sirius Acronically. Riseth ☉ in ♒ 18. Jan. 27.
• Sirius Acronically. Seteth ☉ in ♉ 23. May 3.
• At Athens Sirius Acronically. Ri. ☉ in ♒ 4. Janu. 13.
• At Athens Sirius Acronically. Set. ☉ in ♊ 9. May 20.
• Pleiades Acronically. Riseth ☉ in ♏ 13. Octo. 26.
• Pleiades Acronically. Seteth ☉ in ♉ 29 ½ May 10.
• At Athens Pleiades Acroni. Riseth ☉ in ♏ 19 ½ Nov. 1.
• At Athens Pleiades Acroni. Seteth ☉ in ♉ 27 ½ May 8.
• Arcturus Acronically. Riseth ☉ in ♈0. March 10.
• Arcturus Acronically. Seteth ☉ in ♑ 4 Dec. 15.
• At Athens Arcturus Acroni. Riseth ☉ in ♈ 10. ½ Mar. 20.
• At Athens Arcturus Acroni. Seteth ☉ in ♐ 6. Nove. 18.

CHAP. XXII. To find when a Star riseth or setteth Heliacally.

AStar riseth Heliacally when he geteth out of the beames of the Sun, and beginneth to be seen in the East a lit­tle before Sun rise. And a Star is said to set Helia­cally when he getteth into the beams of the Sun, and beginneth to be least in the evening by reason of the Suns op­proach to him. Those Stars which you see nearest the East Horizon in the Morning Twilight are Heliacall Risers; and those which you see nearest the Westpart of the Horizon, in the even­ing Twilight are Heliacall Setters. For this no exact rule can be given, for all men have not like quickness of sight, nor all Stars like brightness, nor all Climates, Countries and Dayes of the Year the same clearness of Air. And the Moon oft times aug­menteth [Page 87]the Twilight, when she is within a few dayes of the Change, and keepeth the Stars longer Combust. Commonly about twenty dayes before their Acronicall setting they come within the Sun beames, and so set Heliacally, and they appear again, (that is, rise Heliacally) about twenty dayes after their Cosmi­call rising. But if they be great Stars, the Air clear, your sight good, the angle made between the Ecliptique and the Horizon great, they may appear sooner: and later in the contrary Cases. According to this rule the Pleiades set Heliacally, now at A­thens ☉ in ♉ 7. and rise Heliacally ☉ in ♊ 9. so they should be Combust there 32 dayes: but because they be Stars of less Magnitude, we may perhaps allow them 40 dayes as Hesiod did in his time, in the beginning of his Second book of Weeks and Dayes. [...].

CHAP. XXIII. To find the Age when any Astrologer lived, and what time of the Solar year the Seasons hapned in his Country, by knowing his Latitude, and the Rising of any Star in his time.

THe old Grecians, and after them the Latines, before Julius Caesar especially, designed the Seasons of the Year by the rising and setting of some notable Stars. Hesiod begins his second book of Weeks and Dayes: with this Georgical Canon.

That is, when the Pleiades rise, begin to Mow, and to Plow when they set; And in the same Book Vers. 182. he saith

[...]. That is, 60 dayes after the Winter Tropique Arcturus riseth Acronically, and then appears the Swallow, the Spring being then new begun.

These and the like rules were the Husband-mans Almanack. [Page 88]by which they measured the Solar Year, and the return of the Seasons. For in their Civill Year, consisting of Lunar Moneths, by reason of an intercalary Moneth which was added every third Year, and somewhat ofter, the Seasons could happen upon the same day of the moneth yearly, but sometimes 2 or 3 weeks sooner or later, as our moveable Feasts do. The rising and setting of the fixed Stars keep the same distance Yearly, from the Equinocti­al and Solstitiall points, for a mans age near enough; but longer those rules cannot last without some perceivable error: for in 100. years the Stars go forward in Longitude. according to Ty­cho. 1 degree 25 minutes, by reason whereof the risings and set­tings of the Stars happen later in the Year, about a day and half every 100 years in the same Latitude.

Now if you would note in what Age a Star had such a rising or setting, in such a Latitude, as for Example. In what age Arcturus rose 60 dayes after mid-winter in the Latitude of Asera in Boeotia near Athens, whose Latitude is 37 ¼, and con­sequently how long since Hesiod lived, in whose dayes Arcturus had such rising, you shall reason thus: 60. dayes after the Winter Tropick the Sun is in ♓ 1 degree by the Epheme­ris for in those 60 dayes near his Perigium he goeth about 61 degrees. I am therefore to seek when Arcturus did rise at Athens with the opposite degree of the Ecliptique, ♍ 1 degree (the Sun is in ♓ 1 degree, then setting over against it.) I seek the Longitude and Latitude of Arcturus, and find in Tycho'es Tables that Anno Domi. 1600. Arcturus had Longitude ♎ 18. 39 minutes, Latitude B. 31.02. minutes: then I will suppose that Hesiod lived 830. years before Christ (for there some Chronologers place him, but without any good proof that I find) that is, 2430 years before Anno Domi. 1600. in which space of time Arcturus must have increased his Longitude by Tycho' Hypothesis 34 degrees 25 minutes, which being subducted out of the Longitude which Arcturus had Anno Domi. 1600. leaves his Longitude for the year before Christ 830. ♍ 14. degree 14. minutes, his Latitude was then and ever 31.02 minutes. Now from this Longitude and Latitude, I get his Right Ascension, and Declination, by Chap. 34. of this Book, where I find Ascension 180. degrees, Declination North 34. degrees 15 minutes: those had, I place Arcturus in my Reet according to that Right As­cension and Declination (as was taught Book 1 7. and Book 4.18.) [Page 89]and by Chapter 21. I find ♍ 4 rising with him; and at the same time ♓ 4. setting in the same Herizon of Athens. But I ought to find ♓ 1 degree setting in Hesieds time. Therefore I will suppose again that Hesiod lived 1130 years before Christ: and proceeding as upon the former supposition, I find that then ♒ 29. degrees did set at his Acronical rising: but I ought to find ♓ 1 degree rising. And seeing it is hereby found, that in 300 years his Acronicall setting varies 5. degrees, or dayes; I take the proportional part of that time, and lay that in the year 1010. before Christ, Arcturus did set Acronically 60. dayes af­ter the Winter Trepique: and then lived Hosiod, or soon after. For being an Astrologer himself (as Pliny tells us Lib. 18.25. saying, Hu [...]us qua (que) nomine extat Astrologia,) it is likely he would not use an antiquated rule. Arcturus therefore rose Acronically at Athens in Hesiods time, ☉ in ♓ 1 degree, that is, about Febr. 9. of our Julian year, as it now goeth: then the Swallow used to come to Athens: but in our Age he riseth Acronically at Athens ☉ in ♈ 10 ½ that is Mar. 20. and at Ecton or Northampton, ☉ ♈ 0. that is, Mar. 10.

By this you may see that the old Astrological Rules concer­ning the rising and setting of the Stars, left us by Hesiod, Cato, Aratus, Varro, Palladius, Virgil, Ovid Pliny, Columella, Pto­lomy: and other Ancient Authors cannot serve for our Age, nor for every Latitude; and the best use we can make of them, is to find the Age, when they lived.

Pliny, Lib. 18.26. saith, that in Caesurs Calender octavo Ca­lend. Martij was Hirundinis adventus & jostero die Arcturi exor us Vespertimus. Which agrees not to Caesars time.

Also Lib. 2.47. he saith, Ardentissimo aestatis tempore exori­tur Caniculae fidus Sole primem partem Leonis ingrediente, qui dies est 15. ante Caelend. Augusti (that is, July 18.) Rome is in Latitude 42, degrees, Pliny lived about 70. years after Christ; then was Canicula (that is Sarius) in ♊ 17. degrees Latitude, 39 ½, Right Ascension, 79 ½ Declination, South, 16 ⅔, and did rise at Rome Cosmically, decimo quinto Calend. Augusti, or July 18. as thus far he reports truly: but the Sun was not then in ♌ 1 deg. as Pliny saith but in ♋ 23 degrees: for the Sun entred ♌ in his time not decimo quinto Calend, Augusti, but octavo Calend. Augusti. The Sun in those dayes entring the several Signes mostly on the 5 day of the several moneths, as in our Age a­bout [Page 90]the 11 ^th day as Astronomers well know. Pliny seemes to have taken his Astrologie upon trust. And I cannot devise what should lead him to suppose, that howsoever the Equinoxes and Solstices in his time hapned octavo Calend. (as he denyeth not) yet the Sun entred into a new Signe about the Ides of every Moneth, and that the Equinoctial and Soistitial points were in Octavis partibus signorum, as if the Sun came not to the Equi­noctial till he came to the 8 ^th degree of Aries. See Pliny Book 18. Chapter 25, 26, 27, 28. He seemeth to distruct the Julian Calender, and to adhear more to the account used by Varro de Rerusticâ Lib. 1.27. but either he understood neither of them well, or I do not well understand him.

Now Sirius riseth in our Horizon with ♌ 18½, about August 1, in the Declination of the heat, who in Plinyes time rose arden­tissimo astatis tempore. And our Dog-dayes if we follow the Dogs rising will be every age colder and colder, and at length fall in Winter. It were better to reduce them to the Suns entrance into Leo, or to Cancer, 23, rather as they were in Plinyes time: and to count the ardentissimum tempus a fortnight before and a fortnight after: for Sirius was not by the Ancients sup­posed the cause of the sultry heat of Summer but a concomitant signe of that Season, whereof the Suns continuance in the North-Signes was the cause.

Would you know also when they began to Plow and to Mow in Greece in Hesiods Time? He saith, when the Pleiades rise, begin to Mow, and to Plow when they set. The Pleiades (I mean the brightest of them) 1010 years before Christ, were in ♈ 17.25. minutes, Latitude 4 degrees North. Declination therefore by Chapter 34) 11 degrees, Right Ascension 14½ degrees; there­fore they rose Cosmically at Athens or Ascra, ( Hesiods birth Place) ☉ in ♈ 10 ⅓, that is, as our Julian year now goeth, about March 20. The Heliacall rising is about 20. dayes after the Cosmicall (Chapter 22.) that is about April 9. Therefore ei­ther March 20. at the Cosmical rising, or April 9. at the Heli­acall rising, they began to Mow, and I think he means the Cosmicall; the Acronicall rising was there in his Age ☉ in ♎ 10. ⅓ about Sep. 23. which is too late beyond reason. Now that they should begin Mowing in Greece within 10 dayes after the Equinoctial is not strange, seeing the first fruits of ripe Corn were offered at Jerusalem yearly at Easter; which fell ordinarily 15. [Page 91]dayes after the Equinoctial, or thereabout. Duet. 16. And in Egypt - cum falce arva visunt Paulo ante Calendas Apriles, mossis autem peragitur Maio, saith Pliny 18.18. viz. Harvest began in Egypt a little before April, and April then began 8, dayes af­ter the Equinoctial onely.

The Cosmicall setting of the Plaiades at Athens, in Hesiods time 1010. years before Christ was ☉ in ♎ 18. degrees Octob. 1. then began they to Plow and Sow: the Egyptians began No­vembri mense incipiente Pliny 18.18. But if Hesiod were now alive at Ascra he would find the Plaiades rise Cosmically, with ♉ 19 ½ Alpril 30. and set Cosmically ☉ in ♏ 27 ½ Nov. 9. so much are his Georgique rules now antiquated, and serve for little else but to shew how many Ages ago he lived; and how the Seasons hapned in his Age.

CHAP. XXIV. The Latitude of your Place, the Declination, Alti­tude, Azimuth, and Hour of the Sun or Stars, any three of these being given, so find the other two.

SEt your Planisphear in the first Mode of the Meridional Projection, The Comple­mental Trian­gle. and you shall find all these five in one Ob­lique-angled Triangle; which I use to call the Comple­mental Triangle; because it consists of three Sides, which are all Complements. (Others may call it as they please.)

A B in the Limb between the Pole and Zenith, Complement of Latitude.

A C in a Meridian, Complement of the Declination, or the Supplement of that Complement.

B C in an Azimuth, Complement of the Altitude.

A at the Pole is the Angle of Horary distance from the Me­ridian, whose full measure is in the Equinoctial line; but because every Parallel is divided by the Meridians into 180. degrees as the Equator is, and every 5th. and 15th. Meridian plainly distin­guished from the rest in the Fabrique of this Instrument, there­fore you may easily count the angle of the Hour in any Parallel.

B at the Zenith, is the angle of the Azimuth, accounted from the North part of the Meridian: his full measure is in the Finiter­line of the Reet; but you may number it in any Almicanter be­cause [Page 92]every 5th. and 15th. Azimuths are distinguished on the Reet, as the Meridians are on the Mater.

C the place of the Sun or Star, in the meeting of the Meridian and Azimuth, is the third angle, which commonly is neither known nor enquired; but it may be found when you please, by turning the Triangle, as hath been often shewed.

Now if you be versed in the 8 last Chapters of the third Book, you may easily find any of the requisites of this Chapter with­out any more direction. Nevertheless for the Learners sake, I shall exemplifie this general Probleme, in the 4 next Chapters, and also further in the 31, 32, and 33. Chapters hereafter follow­ing. See the Scheam Chap. 26.

CHAP. XXV. To find the Altitude and Azimuth of the Sun or Stars, at a­ny time proposed; the Latitude and Declination being known.

YOur Planisphear set in the first Mode of the Meridional Projection, as in the former Chapter, go to the Parallel of the Declination of the Sun or Star, and follow him through all the Meridians from the Finiter to the Limb; (which is the Meridian of your Place;) and thence back again to the Fini­ter, and you shall find at the first sight what Almicanter and A­zimuth cross the Parallel in any point proposed: and so have you the Altitude and Azimuth thereof.

Example. June 10. the Sun was in the Tropique of Cancer, and so makes his diurnal revolution in the 23 ½ Parallel of Decli­nation; I follow this Parallel, the Tropique, from the Horizon up­wards, and having gone 4 degrees, I meet the ragged arch or hour line of 4. (which is the 120th. Meridian from the South) there crosseth the second Almicantar, and the three and fiftieth Azi­muth from the North; whereby I learn that at 4. in the Mor­ning, June 10, the Sun is 2 degrees high; and in Azimuth from the North 53. Thence going on 15 degrees, I come to the hour circle of 5. where cutteth the 10th. Almicantar almost, and Azimuth 64 degrees, and better: going 15 degrees further I come to the Axtree-line, which is the hour circle of 6. and there I find the Suns Altitude 18 degrees, and his Azimuth from the North 75 degrees, &c. And look what Altitudes and Azi­muths I find at 4, 5, 6. &c, in the Morning, the same I find at [Page 93]the afternoon hours, that have like distance from Noon: because the Eastern and Western Hemisphears are alike, and the same lines serve them both.

Thus you may do in any other Parallel, and for any Star, as well as the Sun; having his Declination given. And so you may make Tables of the Suns Altitude and Azimuth, at every hour, and quarter of an hour, if you please, for every day through­out the year: and that as fast as you can write them, without changing the posture of the Planisphear at all.

CHAP. XXVI. The Latitude, Altitude, and Azimuth given, to find the Declination, and the Hour.

EXample. Having observed the Suns Altitude 13. degrees, and his Azimuth from the South Westward 28 ½ in our Latitude 52 ¼. my Planisphear set in the same man­ner as Chapter 25. I sought out the 13th. Almicantar at the Limb of my Reet, and followed him inwards till I came between Azimuth 28, and 29, there I met the 30th. Meridian, and the 20 ¼ Parallel of Declination, by which I gathered that it was 2 of the clock after noon, and that the Sun declined Southward 20 degrees ¼.

Note here, that the hour of a Star thus found, is not the hour of the Night, unless the Star happen to be opposite to the Sun; but it is the time the Star lacketh to come to the South, or the time of his course from the South.

[Page 94]

CHAP. XXVII. The Latitude, Declination, and Altitude, given, to find the Hour, and Azimuth.

HEre the three sides of the Complemental Triangle are given, and the angles A and B sought.

Example, March the 10 in the Morning the Sun being in the Equinoctial, I observed his Altitude 32 de­grees: the Finiter being set to my Latitude 52 ¼. as before, I went to the 32. Almicantar in the Reet, and where I found him crossing the Equinoctial line of the Mater, there I conclude was the place of the Sun at the time of my observation; and the an­gle C of my Triangle: there the 30 ½ Meridian passing, shewed me that the angle A at the Pole was 30 ½, or, that it wanted half a degree, that is 2 minutes of time, of ten of the clock: and there also the Azimuth 36 ⅔ from the South (or from the North 143 ⅓) shewed me that the angle B at the Zenith is 143 ⅓, the Azimuth from the North, and his supplement 36 ⅔, the Azi­muth from the South.

CHAP. XXVIII. The Declination, Altitude, and Azimuth of the Sun given, to find the Hour, and Latitude.

IN the Meridional Projection, look in the Reet where the Almicantar for the Altitude given, and the Azimuth gi­ven do cross; and turn the point of the Reet where they cross to the Parallel of the Suns Declination upon the Mater: the Meridian that cutteth there sheweth the hour, and between the Finiter and the Pole, or between the Equator and the Zenith, you have the Latitude in the Limb.

CHAP. XXIX. To find the Hour of the Night, by the Northing, or Southing, Rising or Setting of any Star.

USe for this the Equinoctial Projection. And if the Star be in your Reet, turn him to the North or South of the Meridian line, or to the East or West part of the Hori­zon in your Planisphear, as you see him in the Heaven. Then turn the Label to the Suns place in the Ecliptick of the Reet, and it shall shew the hour in the Limb: but if the Star be not in the Reet, you shall supply him by the shift used Chap. 18.

Example. March 10th. I saw Sirius setting in the South-West; and having turned him to the same place in my Plani­sphear, I laid the Label to ♈ 0. which was the Suns place for that day; and it cut in the Limb 11. hours 3 minutes past noon.

Again, December 1. Seeing Lucida Pleiadum in the Meridi­an, I turned the Star till he touched the Meridian line of the Mater; then laying the Label to ♐ 19. the place of the Sun, I found it was 10. hours 17. minutes at Night,

The same Night I saw Ras Aben, or the brightest in the Dragons head under the Pole, in the North part of the Meridian; wherefore I placed him on the Meridian line between the Cen­ter and Septentrio: and the Label laid to ♐ 19. shewed me it was 12. 36. minutes; that is more then half an hour past Mid­night.

CHAP. XXX. The time of Day or Night given, to find in what Coast any Star is: and how much he is distant from the Horizon, or Meridian.

LAy the Suns place to the hour given, in the Equinoctial Projection, then may you presently see all the Stars of the Reet in what Coast they are, whether under the Ho­rizon or above, and how many hours they lack, or are past either the Horizon or Meridian.

Example. Sitting within dores at seven of the clock on Christmas day at Night, I desired to know what Stars were ri­sing, and what near the Meridian, wherefore laying ♑ 14. to 7. of the clock afternoon, I saw in the Reet the Rams horn, a little past South. The Plaiades wanted 1. hour 28. minutes of South, as the Label shewed me in the Limb; ♌ was rising, but Cor ♌. not yet up. I would know now what he wanted of rising, therefore I turned forward the Reet till Cor ♌ came to the Ho­rizon, and observed how many degrees of the Reet passed under the Label (or by any point of the Limb) while the Reet turned: and I found that ♈ 0 (and so any other point) moved on in the Limb 10 degrees in the while that Cor ♌ was comming to the Horizon. Whereupon I understood that he would rise 40 mi­nutes after.

CHAP. XXXI. The Time, and Latitude given, to find the Altitude, and Azimuth of any Star: and thereby to get the know­ledge of the Stars.

FInd the hour of the Star, by the former Chapter. And by the Label observe his Declination: then set your Pla­nisphear in the Meridional Projection, and in the Pa­rallel of the Stars Declination, number his hour-distance from the South; where the number ends, set a needles point; there is the place of the Star; and the Almicantar and Azimuth that cut him there, shew your desire.

[Page 97]Example. I would know the cloudy Star Praesepe, in the breast of Cancer (which indeed is a glimmering light, made up of five smal and bright Stars as by the Telescope appearech) This Praesepe I found by the last Chapter to want 6. hours 21. min. of South; his Declination, found by the Label, is 21. degrees North. Therefore setting the Finiter to our Latitude 52 ¼, I fol­low the 21. Parallel of North Declination, from the Meridian till I come 5 degrees 15 minutes past the Axletree (because I found him 6 hours 21 minutes before the Meridian) there the 17th. Azimuth, from the East Northward, cutteth the Parallel of Praesepe; and there cutteth him also the 13th. Almicantar. Now to find him out, I lay my Planisphear Horizontally, setting the Meridian of my Planisphear in the Meridian of the Place (by Chap. 3, and 4.) and I turn my Label and Sights to the Azi­muth of Praesepe 17 degrees from the East Northward, and be­fore my Sights I hang up a Plumb-line upon a Pole, to keep the Azimuth; then keeping my station, I set my Planisphear upon his edge, or hang him upon a staf with a socket, in the Azimuth of the Star; so that the Plumbet show Altitude 13 degrees (by Chap. 1.) then do the Sights point just upon Praesepe, and would teach me the Star, if I did not know him before.

CHAP. XXXII. The Latitude of the Place, the Declination of a Star, with his Altitude, or Azimuth given, to find both the Hour of the Star, and the Hour of the Night.

BY this Chapter you may find the time of Night, at any time, by any Star, if he be visible above the Horizon. Use the first Mode of the Meridional Projection: and ha­ving Observed the Altitude or Azimuth of the Star, look where that Altitude or Azimuth cutteth the Parallel of the Stars De­clination, there cutteth also a Meridian which sheweth the hour of the Star, that is, the distance of a Star from the Meridian in hours and minutes. And by this hour of the Star, to get the hour of the Night, you shall place the Star at the hour found, in the Equinoctial Projection: which done, the Label laid to the place of the Sun shall shew the hour of the Night in the Limb.

[Page 98]Example. December 25. I observed Procyon to be full East, his Declination North, is 6 degrees 3 minutes. In the Meridi­onal Projection I looked where the Axis of the Reet (which is the East Azimuth) cut the sixth of the North Parallels, and I found the intersection 1 degree South-ward from the Axtree of the Mater (or hour-line of six) there also cutteth the 5 ½ Al­micantar, which shewes more then I sought, that Procyon was 5 degrees 30 minutes above the Horizon. Now having the hour of the Star, 6 hours, 4 minutes, before the Meridian, I take the Equinoctial Projection, and having laid the Label one degree from Oriens South-ward in the Limb, I turn Procyon to the La­bel, which sheweth his Hour 6 hours 4 minutes, and leaving him there, I turn away the Label to ♑ 14 the Suns place, and it shewes me in the Limb the time of night 6 hours 26 min. past noon. And the same I might have found, if instead of his A­zimuth, I had observed his Altitude 5 degree ½, the crossing of that Almicantar which the 6th Parallel would have given me the same hour of the Star, and further, his Azimuth, undesired.

CHAP. XXXIII. Your Latitude known, and the Altitude, and Azimuth, of any Star, Planet, or Comet, observed, and the time of Night: how to find his Right Ascension, and Declination.

THis Case differeth little from the Case of Chapter 26. where, from the same things given, the Declination and hour was required. For the hour and Right Ascension are in a sort the same thing, only the account of the Right Ascension beginneth alwayes at ♈ 0. and is made in de­grees and minutes of a degree. The account of the Hour beginneth at the Meridian, and is made in hours and minutes of an hour. Fifteen whole degrees make an hour, and consequently 15 mi­nutes of a Degree make one minute of Time, for in every minute of Time, there passeth the Meridian a quarter of a degree of the Equinoctial. The time of Night is here further required to be given; which may be had by Chap 29. or 32.

The rule. When you observe the Altitude and Azimuth of the Star, observe also the time of Night, by Chapter 29. or 32. [Page 99]and to save you labour herein, you shall do best to observe your Altitude and Azimuth, when some known Star is seen just in the Meridian. Then with your Latitude, and the Altitude, and Azimuth of the Star, get by Chapter 26. the Declination, and hour of the Star: Then in the Equinoctial Projection, lay the de­gree of the Sun to the hour of the Night: thence turn the Label to the hour of the Star, and you have his Right Ascension in the Limb of the Reet, between ♈ 0 and the Label.

Example. Put case I would find the Declination and Right Ascension of Lucida Pleiadum. The Sun being in Sagittarius 20. December 2. I observed that when Australis caudae Caeti is full South, Lucida Pleiadum was near South-east viz. in the Azimuth 67. from the Meridian, and the Altitude of the said Star 45. 0. hence, by Chapter 26. I find his Declination 23 de­grees North, and the hour-distance from the Meridian 45 degr. that is 3 hours before noon. Then in the Equinoctial Projecti­on (according to Chapter 29.) I set Australis caud. Cati in the Meridian line of the Mater. and turning the Label to 9. of the clock (which is the hour of the Star) I find in the Limb of the Reet (numbring from ♈ 0, to the Label) 52 degrees, the Right Ascension of Lucida Pleiadum: and where the 23 de­grees of the Label now touches the Reet, there may I prick the Star in my Reet, if I have him not before; the time of night is easily seen, by turning the Label to the Suns place, it shewes 7. hours 12 minutes at night: but I need not so much as look on that, though by placing Australis caud. Caeti in the South, I have the time implicitely. The Proposition therefore, might have been thus made; Your Latitude known, and the Altitude and A­zimuth of an unknown Star observed, just at the time when any known Star is in the Meridian; to find both the Right Ascen­sion, and Declination of the Star unknown.

Note also, that if you observe the unknown Star in the Meridi­an Azimuth, you have presently his Declination, by Chapter 13. and the Right Ascension of Culmen Caeli, is the Stars Right As­cension.

CHAP. XXXIIII. The Declination, and Right Ascension of any Star given, to find his Longitude, and Latitude.

LOok the Stars place in the Mater, (which is the inter­section of the Meridian of his Right Ascension with the Parallel of his Declination) and make a prick there. Then your Planisphear being set in the second Mode of the Meridional Projection, you shall presently find the Longitude and Latitude in the Reet: for the Azimuth cutting the said prick, shewes his Longitude, and the Almicantar his Latitude.

Example. November 14. 1639. I observed a Star of the third Magnitude in the Heart of Cetus, which I know to be no common Star, because I had never noted it before, neither could I find it in the Tables of Ptolemy, Tycho, or any other: the Right Ascension thereof was 30. 13 minutes, the Declination 4. 50. minutes South, as I observed by 2 way which hereafter shall be shewed Chapter 44. I made therefore a prick with ink in the Mater of my Brass Planisphear, where the 30th. Meridian (numbred from the Center toward my right hand) and the 5th. Parallel of South Declination do cross; regarding also the odd minutes. Then as soon as my ink was drie, I set the Finitor in the Ecliptique line of the Mater with the Zenith South-wards; because the Latitude of the Star was South, and I saw the 26. Azimuth from the Axtree line cutting the prick, and likewise the 16th. Almicantar cutting about 10 minutes below the prick toward the Finiter. Therefore because in this Mode the Azi­muths be Circles of Longitude, and the Almicantars Parallels of Latitude, (by Book 2.1.) I conclude the Longitude of Cor Caeti. was ♈ 26. and his Latitude 16. 10 minutes South.

When first I observed this strange Star in the said year 1639. and could find no mention of it in the Tables of Ptolemy, Coperni­cus, Stadius, Tycho, or Maginus, I did thereof advertise my very good friends D ^r. John Twysden, then in Kent, and M ^r Samuel Foster, Professor of Astronomy in Gresham Colledge, then at London; who thereupon made the same observation of the Star that I had done, for the place of it; and we all agreed that it in­creased in light, and was above the third Magnitude in December [Page 101]1639. and that it had no perceivable Parallax. And as I was thinking to publish some brief advertisment thereof, in the Latin tongue, that Astronomers beyond the Seas as well as here, might attend the observation thereof, M ^r Foster wrote me word that he had found the Star pictured in Bayerus his Images, which were printed Anno Domi. 1616. And in 1640. there came to me through D ^r Twysdens hands a Treatise of that Star, then newly Printed, by one Phacylides, Professor of Logique at Frane­quers, whose observations agreed with ours. But he thought this Star to have been made of the great Eclipse of the Moon which hapned December 10. 1638. in the foremost foot of ♊. wherein we were not of his mind, you may read this conceipt in his Book pag. 197. This Star doth often appear, and again disappear; it is sometime of the 3d. Magnitude, sometime of the 4th. I have seen it oft in the Eastern Hemisphear, seldome in the Western. It is lost sometimes divers weekes together: this year I could never see it, till February 2. 1656/7. Such as have leasure for the Study of these Arts, may do well to observe it, and to search the reason of its changes: for which purpose I thought it fit to give this notice.

CHAP. XXXV. The Longitude, and Latitude, of any Star given, to find his Right Ascension, and Declination; and to place the Stars in the Mater.

THis Probleme is the converse of the precedent. The Stars are registred by their Longitudes and Latitudes, because their Longitudes increase equally, and their Latitudes remain the same. And so the Tables are easily rectified to any Age, by Addition or Subtraction of a few degrees or minutes of Longitude onely: but the Right Ascen­sion and Declination of the Stars happen to increase and decrease very unequally, and must therefore be calculated from Age to Age, from the Longitude and Latitude whose Tables are more certain.

Set the Planisphear in the second Mode of the Meridional Projection, as in the former Chapter, and bearing in mind that the Azimuths here are Circles of Longitude and the Almican­tars [Page 102]Parallels of Latitude, look where the Longitude and Lati­tude of the Star meet, and there make a prick in the Reet; and look what Meridian and Parallel of the Mater cut, under that prick they shew the Right Ascension, and Declination of the Star.

Example. Eniph. Alpharats, that is, Os Pegasi, had by Tycho'es Tables An. Dom. 1600. Longitude ♒ 26. 22 minutes, Latitude 22.07 ½. North. The Finiter set to the Ecliptique line of the Mater, and the Zenith toward the North Pole (because the Stars Decli­nation is North) I count the Longitude of the Star upon the Fi­niter, (here Ecliptique) thus. At the Center say I, is ♈ 0. thence proceeding rightward to the Limb, I say, here is ♋ 0. whose Right Ascension is 90. thence returning to the Center, I say, here is ♎ 0. upon the Axis of the Reet and Right Ascension 180. upon the Axis of the Mater; thence I proceed in the Finiter to the other side of the Limb, and say here is ♑ 0. bounded by the Limb of the Reet and Right Ascension 270. bounded by the Limb of the Mater, which Limbs here fall into one Circle; and are Colurus Solstitiorum: these numbers I keep, and returning back in the Finiter toward the Center, when I am gone 30 de­grees, I say, here begins ♒. and going on 26. 22. minutes fur­ther, I say, thus far is the Star gone in Longitude. Now here cuts the Finiter (by this account) the Azimuth 35 ⅓. from the Limb; in this Azimuth I number the Stars Latitude, by the Al­micantars 22.07 ½. and at the end of that number in the said A­zimuth I prick the Stars place. And here I see the 8th. Pa­rallel of North Declination upon the Mater cutteth him, and the Meridian 51 ⅓. from the Limb shewing the excess of his Right Ascension above 270. which I kept before. Therefore I conclude the Right Ascension of Eniph. Alpharats, Anno Dom. 1600, was 321. 20 minutes; and his Declination 8. deg. North.

Another Example. In the Year 1670. Aldebaran will have one degree of Longitude more then he had in Anno Domi. 1600. therefore he will be in ♊ 5, 12 minutes Latitude 5. 31 minutes South. Now because the Latitude is South, I turn the Zenith towards the South Pole (the Finiter being placed on the Eclip­tique line as before) and beginning at the Center, I number on the Finiter (here Ecliptique) the Longitude of Aldebaran 65. 12 minutes; and a little beyond the 65th. Azimuth I climbe up by the Almicantars, toward the Zenith 5. 31 minutes to the place [Page]

[Page] [Page 103]of Aldebaran. There the 64 ⅓. Meridian of the Mater cutteth under him; she wing his Right Ascension: and likewise the 16th. Parallel almost of North Declination; shewing that Aldebaran declines North almost 16 degrees, though he have South Lati­tude 5. 31 minutes.

Another way to place the Stars in the Mater by their Decli­nation and Horary-distance from the Meridian. See hereafter Chapter 52.

CHAP. XXXVI. The Latitude, and Declination of a Star given, to find his Longitude, and Right Ascension.

SEt your Planisphear in the second Mode of the Meridional Projection, turning the Zenith Northward or South­ward as the Stars Latitude hapneth to be North or South. Then look where the Parallel of the Stars La­titude in the Reet cutteth the Parallel of the Stars Declination on the Mater, the Azimuth cutting that intersection sheweth the Longitude of the Star; and the Meridian there cutting shew­eth his Right [...]scension.

Example. The Declination of Spica ♍, Anno Dom. 1670. will be 9 ½. South, the Latitude was always 1. 59 minutes South. Now where the second Almicantar cutteth the 9 ½. Parallel of South Declination, there passeth the 19th ¼. Azimuth from the Axis toward my left hand shewing Spica's Longitude ♎ 19 ¼, and the 17th. Meridian from the Axis, to which I add a Semi-circle (because ♎ 0. is at the Center) and I make 197 de­grees the Right Ascension of Spica for 1670.

CHAP. XXXVII. The Longitude, and Latitude of two Stars given, to find their Distance.

MAke one of the Poles of the Mater to be Pole of the Ecliptique, for this turn, and set the Star which hath most Latitude at his distance in the Limb, and turn the Zenith to him; count thence by the Meridians the [Page 104]difference of Longitude, till you come to the other side of your Triangle; and in that side number either the Latitude from the E­quator, or his complement from the Pole; at the end of this num­ber is the other Star: and the Azimuth passing from him to the Zenith, shewes the distance. This is done by the second Pro­bleme of Obliquangled Triangles. Book 3.15.

Example. In Tycho'es Tables for 1600.

Aldebarans Longitude is ♊ 4.12 ½. Latitude 5. 31. min. A.

Sirius Longitude ♋ 8. 35 ½. Latitude 39. 30 ½. A.

Difference of Longitude 34. 23.

I number therefore 39. 30 minutes ½, the Latitude of Sirius from the Equator in the Limb, or the Complement thereof from the Pole, (all is one,) there I set the Zenith to stand for Sirius; then because Aldebaran is distant from Sirius in Longitude 34. 23. minutes, I take the 34 ½, Meridian from the Zenith, and where the 5 ½ Parallel cutteth him, there say I, is Aldebaran (and C of my Triangle) and the Azimuth passing thence to the Zenith measureth the distance of the Stars 46 degrees almost.

CHAP. XXXVIII. The Declination, and Right Ascension of any two Stars given, to find their distance.

DO here with the Right Ascension and Declination as you should do with the Longitude and Latitude, by the former Chapter, for the case is like, and requireth the same manner of working.

CHAP. XXXIX. The Declination of a Star or Planet, and his distance from a known Star given, to find his Right Ascension.

BEcause this Case is the converse of the precedent, and so­luble by the first Probleme of Obliquangled Triangles, Book 3. 14. an Example, or two shall suffice.

Anno Domi. 1639. I observed the Declination of Cor Caeti (the strange Star mentioned Chapter 34.) to be 4. 50 minutes South; and his distance from Lucida Mandibulae Caeti [Page]

[Page 105]to be 13.04 minutes and Lucida Mandibulae was Eastward from him. The Right Ascension of Lucida Mandibulae then was 40. 56 minutes, his Declination North 2. 40 minutes; there­fore I have a Triangle whose Sides are all known.

A B the distance of Mandibulae from the Pole 87. 20 mi­nutes, I set between the Pole and Nadir in the Limb, because B C will reach beyond the Finitor.

For A C the distance of the Stars. I seek the 13th. Parallel from the Pole. And

For B C I seek the 94. 50 minutes Almicantar, counted from the Nadir (that is the 5th. almost above the Finitor) and where the said Parallel and Almicantar cross, there is Cor Caeti, and C of my Triangle: through it there cutteth the Azimuth 10 ⅔, shewing the Difference of the Right Ascension of the Stars; which difference I subtract out of the Right Ascension of Man­dibula, because he was further East; and there remaineth the Right Ascension of Cor Caeti 30. 16 minutes, or rather 13 minutes. And I have here also numbred by the Meridians, the angle A at Mandibula 120 degr. though un-required.

Another Example. January 7. 1656/7, I observed by my Brass Quadrant of 12 inches in Radius, the Meridian Altitude of Jupiter 56. 20 minutes, out of which subtracting the height of the Equator here at Ecton 37. 45 minutes; I found his Declina­tion 18. 35 minutes North; his distance then from Lucida Plei­adum, I observed by my Cross-staff 5. 12 minutes, and from Al­debaran 10.07 minutes.

The Complement of Decli. of Lucida Pleiadum is 67.00 mi.

The Complement of ♃ his Declination was observed 71. 25.

And these two Complements with the distance of ♃ and Luci­da Pleiadum 5. 12 minutes, make a Triangle, soluble by the first Probleme of Obliquangled Triangles; whereby you may find the angle of the difference of Right Ascension of Lucida Pleia­dum and ♃ is 2. 56 minutes; which added to the Right Ascensi­on of Lucida Pleiadum (because ♃ was East-ward) maketh 54. 44 minutes the Right Ascension of Jupiter.

CHAP. XL. The Latitude of a Star or Planet, and his distance from a known Star given, to find his Longitude.

DO here with the Longitude and Latitude as you were taught to do with the Right Ascension and Declination, in the former Chapter.

CHAP. XLI. To find the distance of two Stars by their Altitudes, and their difference of Azimuth observed at the same time.

THe Complements of the Altitudes are the distances of the Stars from the Zenith: Set one of the Stars at the Pole, and set the Zenith as much from him in the Limb as the Complement of his Altitude comes to, then considering what difference of Azimuth the Stars had, take the Azimuth of like distance from the Limb (beginning from that side of the Limb where the Pole aforesaid is) and in that Azi­muth reckon from the Finitor the Altitude of the other Star (or the Complement of his Altitude from the Zenith, all is one) at the end thereof is C, and the other Star; and the Meridian that passeth from him to the Pole, shewes the distance of the Stars. This case is so like that of Chapter 37. that he who knowes one may know the other also.

CHAP. XLII. To find the Angles of Station which any two Stars make with the Pole, by their Right Ascension and Decli­nation: or with the Pole of the Ecliptique, by their Longitude and Latitude: or with the Zenith, by their Altitude and Azimuth.

THis Case agrees with the second Probleme of Oblique-angled Triangles.

Example. In the Triangle of Chapter 39. made between [Page 107]the Pole of the World, Mandibula Caeti, and Cor Caeti, I would know the angle at Mandibula, which is the angle of his Station. Place the Triangle upon your Planisphear as in Chapter 39. where the angle unsought, there discovered it self to be 125. degrees.

CHAP. XLIII. To find whether three Stars be in one great Circle, by ha­ving their Longitude and Latitude, or their Right Ascension and Declination, or their Azimuth and Altitude known.

EXample. I would know whether the three Stars of Orions Girdle be in the same great Circle. Here I prick them down, and draw their Circles of Longitude to meet at the Pole of the Ecliptique; so have you two Triangles joyned in one, and the three Stars in the Base of it.

Now first, I must find by the former Chapter what angle of station the first Star hath in the little Triangle P A B, and then what angle of station he hath in the whole Triangle P A C, and if these two angles be equal, then be the Stars all in one great Circle, otherwise not.

This Probleme may be of use to find how the tayle of a Comet pointeth upon the Sun, or upon any other Planet or Star, below the Horizon. But if the three points enquired of, be all in view; I know no better way then to stretch a thrid straight at a reasonable distance from your eye, applying it to the Stars; for if the same straight line cut them all, they be all in one great Circle, otherwise not.

CHAP. XLIV. If a Comet or Star unknown be seen in a straight line with two other known Stars, and his distance from one of the known Stars be observed; how to find the true place of the Comet or Star unknown.

EXample. Anno Domi. 1639. I obser­ved that Mandibula Caeti, Gene Caeti,

and the strange Star Cor Caeti (spo­ken of Chapter 34.) made a straight line; and by a Radius, such as I then had at hand, I observed that Cor Caeti was distant from Mandibula 13. 04 minutes. Now if I would find the Longitude and Latitude of Cor Ceti, I should use the Longitude and La­titude of the other Stars; but because I in­tend to find first his Right Ascension and De­clination, I make use of their Right Ascension and Declination. The manner of working is alike. The Scheme of the last Chapter may serve here if you turn it up-side down; Then P is the North Pole, A Mandibula, B Gena, C Cor Ceti. P A is the Com­plement of Declination of Mandibula 87. 20 minutes, P B the distance of Gena from the North Pole 91. 16 minutes, (for he declines Southward 1. 16 minutes) A P B the difference of their Right Ascension 5. 35 minutes.

Therefore I set the Nadir of my Reet as far from the Pole as P is from A, and so between them on the Limb is the side P A. Then for the side P B it reacheth from Nadir beyond the Finitor 1. 16 minutes, therefore in the 1 ¼. Almicantar I number from the Limb 5. 35 minutes, the difference of Right Ascensions for the Angle A P B, and where the 1 ¼. Almicantar and the 5 ½. Azimuth do meet, there is B for Gena: thence I go in a Meridian to the Pole at A, and as I go I number the distance of B and A, that is, Gena and Mandibula, 7 degrees almost; and I observe that this Meridian is the 125. Meridian from the Limb; so much is A, the angle of station at Mandibula.

[Page 109]Now I say, in this Meridian also is Cor Ceti, because he is in a right line with the other two Stars which are cut by this Meri­dian: and he is 13. 4 minutes from Mandibula, by observation; therefore I run from the Pole A so many degrees in this Meridi­an, and so come to C, the place of Cor Ceti, and there cutteth the 4. 50 minutes Almicantar, shewing the Declination of it, and the Azimuth 10. 43 minutes; shewing his difference of Ascension from Mandibula; which difference I subduct from the Right As­cension of Mandibula (because Mandibula is further East,) and there remains 30. 13 minutes, the Right Ascension of Cor Ceti: which being found, you may find his Longitude 26 degrees, La­titude 16. 10 minutes, by the 34th. Chapter: Observe how your Triangle lies in the Planisphear, where Nadir is used for the North Pole, the North Pole is the place of Mandibula, and the 125. Meridian represents the great Circle cutting the three Stars.

CHAP. XLV. The distance of a Planet from two known Stars being Observed, to find his Longitude and Latitude.

IT is true that M ^r. Blagrave saith, Book 5. 25. that in Questions of this sort it is harder to conceive how they should be resolved, then to resolve them. And therefore he adviseth to draw a rude Scheme of your work, agreeable to the Meridional Projection of your Planisphear, after this manner.

December the 28. 1656. I observed somewhat grosly by my Cross-staff that ♃ was between the Hyades and the Pleiades, distant from Aldebaran 9. 49 minutes, and from Lucida Plei­adum 5.26. and to the Southward of the Stars. I draw there­fore a rude Scheam representing somewhat near the posture of these three Stars. E C is Ecliptique, and P his South Pole, P C the Circle of Longitude of the Westerly Star Lucida Pleia­dum ♉ 25. 12 minutes, and because he hath North Latitude 4 degrees, I place him at F; Aldebaran, whose Longitude is ♊ 5. Latitude South 5. 31 minutes, I place somewhat like at O, and Jupiter I place below the line drawn between them, and nearer to the Pleiades then to Aldebaran, as I observed his situation in the Heaven.

[Page 110]

That I seek now here, is P ♃, the complement of Jupiters La­titude; and F P ♃, his difference of Longitude from Lucida Plei­adum.

First, in the great Triangle F P O, I have the angle P, the difference of Longitude between Lucida Pleiadum and Alde­baran 9 degrees 48 minutes, and the including sides P O 84.29. ( Aldebaran distance from the South Pole) and P F 94. (distance of the Pleiades from the South Pole) and hence by the second Problemes of Obliquangled Triangles Book 3.15. I get at once, the Base O F, distance of Lucida Pleiadum and Aldebar an 13. 45 minutes, and the angle of station at F, viz. P F O 45. 28 minutes.

2. Then in the Triangle F O ♃, whose three sides are now known, I get the angle O F ♃ (by the first Probleme of Ob­lique Triangles, Book 3.14.) 35. 14 minutes) which be­ing Subducted from the angle O F P, leaveth the angle P F ♃ 10. 14 minutes.

3. In the Triangle P F ♃, having now the angle F, and the sides including it, I get the third side P ♃, the Complement of ♃ Latitude 88. 39 minutes (by Oblique Problemes 2. Book 3.15.)

[Page 111]And lastly the three sides in the Triangle P F ♃ being now known my Planisphear unmoved will shew me F P ♃ 58 mi­nutes (by the Probleme 1 Oblique Triangles,) which 58 minutes being added to the Longitude of Lucida Pleiadum maketh up the Longitude of ♃ ♉ 26. 10 minutes, and his Latitude was even now found 1. 21 minutes South.

CHAP. XLVI. To find the Culmen Caeli, and the Altitude thereof, at any time proposed.

CUlmen Caeli is the degree of the Ecliptique which is cut by the Meridian of your Place.

Use the Equinoctial Projection, where having laid the place of the Sun to the Hour proposed, look what degree of the Ecliptique is cut by the Meridian line, and you may number his Altitude from your proper Horizon.

Example. March 29. 1652. I laid the Suns place ♈ 19. 11 minutes to 32 minutes past 10. of the Clock before noon, and in the Meridian I saw ♓ 25 ½ Culminating. And for his Alti­tude I looked where my Horizon cuts the South part of the Meridian (at 52 ¼ from the Center) and from that cutting I count in the Meridian to the Ecliptique 36 degrees, the Altitude of Culmen Caeli.

But note, That if ♓ had been a North Signe, I must have counted first to the Limb 37. 45 minutes, and thence back again to Culmen 1 ¾. in 39 ½.

CHAP. XLVII. To find the Ascendent or Horoscope, and the other three Principal Houses, for any time proposed.

A Strologers divide the Heaven into twelve Houses, of which, four are principal. The First House, which be­ginneth at the Ascendent or Rising point of the Eclip­tique. The Fourth, which beginneth at Imum Caeli, or Midnight. The Seventh, which beginneth at the Descendent point of the Ecliptique. And the Tenth, which beginneth at Medium Caeli, or Culmen.

[Page 112]These be the four Cardinal points, and the Ascendent and Descendent, and likewise the Medium and Imum Caeli, are al­wayes opposite one to the other, so that one being known, the o­ther is known also.

To find these points, use the Equinoctial Projection, and there lay the Suns place to the hour proposed: then the degree of the Ecliptique rising in your Horizon is Ascendent, and you shall see the same degree of the opposite Signe Descending in the West part of the Horizon; and look what degree toucheth the South part of the Meridian, that is, Medium Caeli, and the same degree of the opposite Signe shall be in Imo Caeli, that is, in the North and Subterranean part of the Meridian.

Example. March 29. 1652. I observed the great Eclipse of the Sun, the middle whereof hapned at Ecton, at 10 hours 32 minutes 04 seconds before noon in apparent time, at what time the Sun was darkned digits 11. 22 ½, in ♈ 19. 11 minutes. I would know for this time the Figure of the Heavens.

Therefore laying the Label to 10.32 minutes before noon, and bringing ♈ 19. 11 minutes to the Label, I see in our Horizon ♋ 24. 7 minutes rising, and ♑ 24.7 minutes setting. In the Me­ridian above the Horizon I see ♓ 25. 19 minutes: and in Imo Caeli, toward Septentrio, ♍ 25. 19 minutes.

CHAP. XLVIII. To find the beginnings of the other eight Houses.

THere be six great Circles, by which the twelve Houses are distinguished. They be called Circles of Positi­on: and so they call the rest of the Circles which serve to subdivide the Houses. They be all Hori­zons to some Country or other in the World, and therefore are most fitly represented by the Horizons of the Mater. The First House beginneth alwayes at the Ascendent: and the rest follow in order according to the Sequel of the Signes. But Astrolo­gers are not well agreed about their situation. For 1. Some will have the domifying Circles drawn from the Poles of the E­cliptique through every 30 ^th. degree thereof, as Ptolemie. 2. Some draw them from the Poles of the World, through every 30 ^th. [Page 113]degree of the Equator; as Alcabitius. 3. Some draw them from the intersections of the Meridian and Horizon, through e­very 30th. degree of the Equator; as Regiomontanus. 4. Some draw them from the same intersections, by every 30th. degree of the Prime Vertical or East Azimuth; as Campanus. Yet e­very Astrologer will pretend he can tell you your Fortune, though they go about it so divers wayes, that they may be all false, and but one of them can be true: and no Man hath shewed any bet­ter reason for his way then another, but his own opinion.

If you will follow the first or second way, the matter is plain. For in the first way every 30th Circle of Longitude reckoned from the Ascendent downwards, and so round, is a Domifying Circle: and likewise every 30th. Meridian from the Ascen­dent is a Domifying Circle in the Second way. And if you know but what Longitude a Star hath, you presently find in what House he is, after the first way. And if you know the Right Ascension of a Star, and of the Ascendent, you presently find in what House he is, after the second way.

But if you will use the third way (now commonly used) you shall set the Zonith line of the Reet to the Latitude, and so the Azimuths are your Circles of Position; then look what Azi­muth cutteth every 30th degree of the Equinoctial, that is a Domifying Circle; and you shall reckon here from the Limb, which shall stand for the beginning of the 10th. House, and so in our Horizon 52 ¼. the fourty third Azimuth cutteth the 30th. degree of the Equator, serving the 11th. and third House: and the Azimuth 70 ½ cutteth the 60th. degree of the Equator, serving the 12th and second Houses: and because I know that on the other side the Center the Intersections will be like, I look no further.

But now I must get the Depressions of these Circles under the Pole in this manner. I number in the fourty third Azimuth the Latitude of my Place from the Zenith; to the end of which number I lay the Label, and I see the Azimuth cutting on the Label 32 ½ for the depression of that Circle. And in like man­ner, laying the Label upon the 52 degrees ¼ of the Azimuth 70 ½ I find on the Label his depression 48 degrees; by the third Pro­bleme of Rectangl. Trangles, and the third Variety Book 3.5.1 [...].

These Horizons therefore I choose out in the Mater, viz. 32 ½, and 48. for these with the Meridian and Horizon of my Place, [Page 114]shall serve to get the Houses for ever, in my Latitude: for the 32 ½ Horizon shall be the beginning of the 11th. and third Houses; and the 48th, the beginning of the 12th and Second, Thus have I the Circles of Position of the 11, 12, 2, and 3. Houses: and the 10th. and first, are had by the former Chapter: and these six being had, I have all; for opposite Hemisphears are alwayes alike, and one description serveth both.

CHAP. XLIX. To know what degree of the Ecliptique is in the begin­ing of every House.

DO as in this Example. By Chap. 47. I had the degree Cul­minating in the middle of the great Eclipse there men­tioned, ♓ 25 ½. First I lay ♓ 25 ½ to the Axtree line at 6 in the morning, where it lieth as in a Right Horizon; thence I move it 30 degr. South ward in the Limb, viz. to 8. of the clock, and in the Horizon of the 11th. house (32 ½) I see ♉ 7 de­grees setting, the said degrees of Culmination: 30 degrees further, viz. to 10. a clock, I see in the Horizon of the 12th. House (48.) ♊ 24 ½. And setting the said degree of Culmination to the Noon-line, I see in our Horizon (52 ¼ which begins the first House) ♋ 24 ½ ascending. And setting the said degree 2 hours further on, I see in the Horizon of the second House (48) ♌ 13. And setting the said degree to 4 a clock, I see in the Ho­rizon of the third House (the 32 ½) ♍ 1.

Thus have I the degrees of the Ecliptique in the beginning of 6 Houses, and the 6 Houses opposxe begin with the same de­grees of the opposite Signes.

CHAP. L. Another way to find what degree of the Ecliptique is in the beginning of every House, and thereby to set a Fi­gure more easily then by the former Chapter.

IT was found by Chapter 48. that the Azimuths 43. and 70 ½ are evermore Domifying Circles in our Latitude (52 ¼.) and how you may find them for any other Latitude was [Page 115]there shewed. There I reckoned them from the Limb; but here I shall reckon them from the Axis; and say, the Azimuth 47.

serveth the 11th. House; the 19 ½ serveth the 12th. the Axis for the first, and seventh; the 19 ½ below the Axis for the second; the 47th. for the third; the Meridian for the 10th. and 4th. You may therefore mark the ends of these A­zimuths in the Fini­tor, setting 12 to the 19 ½ above the Axis, (here made Horizon for this turn) and 11. at the 47th. at the Axis, 1. then at the 19 ½ below the Axis, write 2. at the 47th. write 3. and at the Limb 4.

Your Houses being thus distinguished on the Reet, get the degree of Culmination, and the Altitude thereof, by Chapter 46. then set the Zenith under the North Pole, so much as the Alti­tude of the Culmination comes to: and if the Ascendent be a North Signe, let the Pole be toward your left Hand; and con­trary if it be a South Signe: so shall the Axis of the Reet be Ho­rizon, and the Pole Culmen Caeli.

Next get the Ascendent by the 47th. and his Amplitude by the 16th: this Amplitude you shall number in the Axletree of the Reet from the Center alwayes to your left Hand, or toward Septentrio; and mark what Meridian there cuts the Axletree of the Reet, in that degree of Amplitude; that Meridian shall be your Ecliptique for this time: follow him up to the Pole, and you trace out the arch of the Ecliptique from the Ascendent to mid-heaven: and if you go down in his match to the like degree of Amplitude on the other side of the Center, there is the West­ern arch of the Ecliptique from the mid-heaven to the Descen­dent; and here you may see every degree of the Ecliptique a­bove the Horizon, and in what House it is, without any more coursing after them.

[Page 116]Example. ♓ 25 ½ was Culminating; his Altitude 36 degrees; the Ascendent had ♋ 24. whose Amplitude is 36 ⅓. Setting the Zenith therefore 36 degrees to the right Hand under the Pole, I number in the Axtree-line of the Reet, from the Center to my left Hand the Amplitude of the Ascendent 36 ⅓. there cometh the 23. Meridian from the Center, who must serve for the Eclip­tique. Now because it is troublesome to number the degrees of the Signes backward, I will begin at the Descendent 36 ⅓ from the Center on the other side: and say, Here is ♑ 24 degrees descending, (because ♋ 24. was ascending,) hence I count on toward the Culmen, till I come to the Azimuth 19 ½, (which is the Domifyer of the 12th. and 8th. Houses,) and here I say, be­gins the 8th. House in ♒ 13. for there are but 19 degrees from the Descendent hither: hence I count to the 47th. Azimuth (the Domifyer of the 9th. and 11th. Houses) and there I count ♓ 1 degree, for the beginning of the 9th. House: hence I number on to the Pole, and there I happen on ♓ 25 ½, the Culmen and be­ginning of the 10th. House. Thence I number on the other side of the Maters Axtree, in the twenty third Meridian, toward the Ascendent, and I find the 47th. Azimuth cuts ♉ 7 degrees for the beginning of the 11th. House; but the 19 ½ Azimuth which should shew me the 12th. House, is cut off by the Finitor, and I am left to seek him else where: And to find him I need but turn about my whole Planisphear (the Reet unmoved) and make the other Pole Culmen for this turn, and then I find among the A­zimuths that peece of my Ecliptique which I wanted in the for­mer posture; and I may reckon on him between the Ascendent and the Azimuth 19 ½, 29 ½. and thereby see that ♊ 24 ½ is in the beginning of the 12th. House: so have I 6. of my Houses, and may by them find the other 6, (as was shewed Chapter 48.) and set them down as in the Figure.

CHAP. LI. A third way to set a Figure with less labour.

LEt the Meridians and Azimuths here change their offices in which they served in the former Chapter: that is, let the 19 ½ and 47th. Meridian on both sides the Axis of the Mater be Domifyers; and let the 23 Azi­muth [Page 117]be Ecliptique: and to that purpose, set the Zenith above the Pole, according to the Altitude of Culmen 36 degrees, and make the Axis of the Mater Horizon. Then beginning as you did before at the Descendent, go up in the Ecliptique till you come to the Meridian 19 ½, and follow the Almicantar that there cut­teth to the Limb, and there make a mark for the 8th House; then mark where the same Ecliptique cuts the next Domifyer (the 47th. Meridian) and follow the Alm̄icantar from that point to the Limb; prick there the 9th. House: the Zenith is the 10th. thence go toward the Ascendent, and do in like manner; making pricks for the 11th. and 12th. Houses: also in the Limb of the Reet at the end of that Almicantar which cutteth the beginning of the Houses in the Ecliptique. Then in the Zodiaque of the Ring, look the degree of Culmination, and set the Zenith of the Reet to it; and the Label laid to these pricks, shall shew you pre­sently in the Zodiaque the degrees for the beginning of every House.

CHAP. LII. How to place any Star or Planet in his proper House.

IN the Equinoctial Projection, get the Stars Hour distance from the Meridian, thus. Lay the Suns place to the hour proposed: then turn the Label to the Star (or to his Right Ascension if he be not in the Reet) and it shall shew in the Limb how many hours and minutes the Star is past or short of the Meridian: get also the Stars Declination North or South, by the Reet, or by the 35. or some other Chapter; and where the Parallel of the Stars Declination crosseth the hour of the Star in the Mater, there is his place for this turn: therefore having made a prick with ink for him there, set the Zenith line to your Latitude, and having your Domifying Azimuths marked upon the Reet (as Chapter 49. 30.) you shall presently see in what House the Star is.

Example. 1652. March 29, 10. hours 32 minutes before noon, I would know in what House the Pleiades are. The hour of Lucida Pleiadum for that time is 8. 14 minutes after midnight: the Declination is 23 degrees North. I number therefore in the 23. Parallel of North Declination from the Atree [Page 118]of the Mater to the Meridian 33 ½, there is the place of Lucida Pleiadum, where I prick him down; and setting the Zenith line to the Latitude, I find the 39 Azimuth or Circle of Position cuts him: by which I see he is 8 degrees from the beginning of the 11th. House, for that begins at Azimuth 47, as appeares Chap­ters 49, 50.

CHAP. LIII. To find the division of the Houses, according to Campanus.

CAmpanus begins the Houses at every 30th. degree of the East Azimuth, accounting from the Ascendent in the Sequel of the Signes, as was said Chapter 48. Therefore if you will use his way, set the Zenith line to the Latitude, and the Finitor shall become the East Azimuth; and every 30th, Azimuth from the Limb, or Axtree line, is a Domifying Circle: you shall therefore in stead of Azimuths 19 ½. and 47. (which are Domifyers after Regiomontanus for our Latitude, as was shewed Chapters 49, 50.) take Azimuths 30. and 60. on both sides the Axtree line, which are distinguished to your hand; and with these Domifyers you shall work in all res­pects as you did with the other in the three former Chapters.

CHAP. LIIII. How to Direct a Figure.

TO Direct is to turn on the Sphear, till some Star in the second House come into the first, or contrarily: and so observe how many degrees the Equinoctial is moved forward or backward in the same time.

Place the Star or Planet on the Reet, or on the Label, (as Chapter 18. is taught, then in the Equinoctial Projection (as Chapter 47.) set the Ascendent at your Horizon, and note the degree Culminating: then turn on the Reet forward or back­ward till the Star come to the Horizon: then lay the Label on the degree which Culminated before; and mark how many degrees of the Limb he is distant from noon, so many degrees of the E­quator have passed the Meridian and Horizon: which Astrolo­gers take to signifie so many years before the effect promised by the said Star shall happen.

[Page]

CHAP. LV. To find the Angles of the Ascendent, or the Angle of the Ecliptique with the Horizon, and the Altitude of the Nonagesimus gradus, at any time.

IN the Equinoctial Projection set the Suns place to the time proposed, and get the Altitude of Culmen Caeli, by Chapter 46. Then, if the Eastern arch of the Ecliptique be shorter then the Western, you shall count the degrees between the Ascendent and Mid-heaven; otherwise count from the Defcen­dent to Mid-heaven. Number these degrees on the Label from the Center, and where they end make a prick; which prick if you put upon the Parallel of the Altitude of Culmen Caeli, you shall have in the Limb, between the Finitor and the Label, the measure of the lesser angle, which taken out of 180 degrees lea­veth the greater angle. This is done by Probleme 2. Rectang. And note that the lesser angle, and the Altitude of the Nonagesi­mus gradus be alwayes equal.

Example. March 29. 1652. 10. hours 32. minutes a. m. ♓ 25 ½ was in Culmine: the Meridian Altitude thereof is 36. Between Culmen and the Descendent I find 61 ½. therefore I prick the degree 61 ½ from the Center in the Label; and when I have turned that prick to the 36. Almicantar, the Label shewes in the Limb of the Reet 42 degrees for the lesser angle of the E­cliptique with the Horizon (exactly 41. 58 minutes) which al­so is the Altitude of the Nonagesimus gradus: the greater angle is 138 degrees 2 minutes.

Another way. In the Equinoctial Projection: lay the Label on the Nonagesimus gradus; and observe his Declination on the Label, and his Horary distance from the Meridian. Then in the Meridional Projection and his first mode, observe where that Declination and Horary distance meet on the Mater, and the Almicantar touching the same point sheweth the Altitude of No­nagesimus gradus, which is equal to the angle sought.

Example. In the former Case, where ♓ 25 ½ was in our Me­ridian, ♈ 24. was Nonagesimus gradus; the Label laid to it shewed me his Declination 9 ½ almost, North: and his Horary [Page 125]distance from the Meridian in the Limb 26. 20 minutes. then the Fimtor being set to the Latitude, I seek the Intersection of the 9 ½ Parallel of North Declination with the 26 ⅓ Meridian from the Limb: and there toucheth the 42 Almicantar; shewing the Al­titude of Nonagesimus gradus, and the quantity of the lesser angle sought, as before. And there cometh unasked also, the 36 ⅓ Azimuth being the Azmuth of the Nonagesimus gradus, which is alwayes equal to the Amplitude of the Ascendent. O­ther wayes, See Chapter 56. and 57.

CHAP. LVI. The Ascendent and his Amplitude, and the Altitude of Culmen Caeli given; so to represent the Ecliptique, that you may presently find not onely the Altitude of the Nonagesimus gradus, but the Altitude and Azimuth of every degree of the Eclip­tique at one view.

SEt your Planisphear in the third Mode of the Meridional Projection: that is, If the Ascendent be a North Signe, move the Finitor from Meridies toward the North Pole, till the North Pole be elevated above the Finitor according to the elevation of Culmen Caeli; but if the Ascendent be a South Signe, move the other end of the Finitor from Septen­trio toward the North Pole, till the Pole have the Elevation of Culmen Caeli. Then number the Amplitude of the Ascendent upon the Finitor from the Center to your left hand, (toward Septentrio) and take the Meridian that crosseth there for the Eastern arch of the Ecliptique, and his match so much distant from the Axtree towards Meridies shall be the Western arch; so do the Azimuths and Almicantars of the Reet shew at once the Altitude and Azimuth of every degree of the Eclip­tique.

Example, March 29. 1652. 10. hours 32 min. a. m. I found ♓ 25 ½ Culminating. and his Meridian Altitude (by Chapter 46) 36 degrees, the Ascendent ♋ 24. (by Chapter 47.) and his Amplitude 36 ⅓ (by Chapter 15, and 16.) the Sun being then Eclipsed in ♈ 19. 11 minutes. I would know his Altitude and [Page 126] Azimuth, and likewise the Altitude and Azimuth of the Nona­gesimus gradus. To this purpose, I take the North Pole for Culmen, and set the Finitor 36. below him toward Meridies; and from the Center toward my left hand, I number on the Fi­nitor the Amplitude of the Ascendent 36 ⅓, there cuts the twenty third Meridian from the Axis: (which here serveth for the Eastern arch of the Ecliptique:) the degree in this Ecliptique here cut by the Finitor is the Ascendent ♋ 24. thence I number in this Ecliptique South-ward 90 degrees by help of the Parallels, and so I come to ♈ 24 degrees, being the Nonagesimus gradus. Here the 42 Almicantar toucheth the Nonagesimus gradus, shewing the Altitude thereof, and here also cutteth the Azimuth of the Nonagesimus gradus 36 ⅓ equal to the Amplitude of the Ascen­dent, as it is alwayes and ought to be; so as that you might have found the Nonagesimus gradus by this Azimuth with less num­bring. Now for the Sun, he is in ♈ 19. 11 minutes, that is, nearer the Meridian then the Nonagesimus gradus by almost 5. degrees: I count therefore 4. 49 minutes (for so it is) past the No­nagesimus gradus; there is the Sun, and the Almicantar cutting there shewes his Altitude 41 ⅔, and his Azimuth is shewn by the 29 Azimuth some what near. Or if you would reckon after the order of the Signes, which is easier, begin at the Descendent, where is ♑ 24. thence 61 ½ makes ♓ 25 ½ at the Pole, for Cul­men Caeli: thence in the Eastern arch to the Suns place I make 85 degrees 11 minutes; and 4.49 minutes further is ♈ 24. the Nonagesimus gradus.

CHAP. LVII. To do the same another way, by the Horizontal Pro­jection, very plainly.

TAke the Zenith for the Ascendent, and set him in his place in the Limb (which here is Horizon) so much from Oriens as his Amplitude comes to; and that toward Septentrio, if it be a Northern Signe, or if it be a Southern Signe toward Meridies. Then number upon the Meridian line from the Limb inwards the Altitude of Culmen Caeli, and the Azimuth that cutteth there shall be your Eclip­tique in this Case: If the Azimuths reach not the Meridian, [Page 127]turn about the Reet, and set Nadir for Ascendent. Lay the Label to any degree of this Ecliptique, and the degrees of the Label from that degree to the Limb shall be the Altitude thereof: and between the Label and Meridies in the Limb the Azi­muth thereof.

Example. Because in the Case of the former Chapter I foresee that the Sun will be past the Nonagesimus gradus, and so in the West Quadrant of the Ecliptique (though he be in the East Quadrant of the Horizon) therefore I set Nadir at the Amplitude of the Ascendent viz. 36 ⅓ from Oriens North-ward, then in the Meridian line I number from the Limb inwards 36. for the Altitude of Culmen, where I make a prick, and say, Here is ♓ 25 ½ Culminating, and through that prick passeth the 42. Azimuth from the Limb, which is now my Ecliptique; and by that I see that the angle of the Ecliptique which the Horizon (called the angle of the Ascendent, and alwayes equal to the Altitude of Nonagesimus gradus, as was said) is 42 degrees: and if I follow this Azimuth to the Finitor, there is Nonagesimus gradus, and the Altitude thereof 42 degrees counted from the Limb (here Horizon) the Azimuth thereof lies in the Limb be­tween the Finitor and the Meridian 36 ⅓ as before, equal to the Amplitude of the Ascendent, I number also from ♓ 25 ½ in the Meridian 23. 41 minutes to the left hand still, and there I have ♈ 19. 11 minutes, the Suns place, which cuts on the Label 41 ⅔. for the Altitude of the Sun there, and the Label at the same time cutteth in the Limb about 29. from South East-ward for the Azimuth of the Sun: and after the same manner you have before you the Altitude and Azimuth of every other degree of the E­cliptique for the time proposed.

CHAP. LVIII. To do the same by the Nonagesimal Projection, if the Altitude of Nonagesimus gradus be first given in­stead of the Altitude of Culmen Caeli.

SEt your Planisphear in the Nonagesimal Projection (by Book 2.3.) that is, make the Limb now to represent the Circle of Longitude or Azimuth (for it is both) which [Page 128]cutteth the Nonagesimus gradus, and make the Equinoctial line here to be Horizon: and from the Equinectial line number in the Limb the Altitude of Nonagesimus gradus, and thereto set the Finitor, so shall the Finitor be Ecliptique, the Nonagesimus gradus at the Limb, the Ascendent and Descendent at the Center: and because the Equinoctial line is Horizon in this Projection, therefore the Meridians become Azimuths, and the Parallels Almicantars, shewing the Altitude and Azimuth of every degree of the Ecliptique, if you reckon as you ought in this manner. Reckon in the Equinoctial line (here Horizon) from the Center the Amplitude of the Ascendent, to the right Hand, if it be a North Signe, and contrarily if it be a South Signe. Where this Amplitude ends is the East point, from whence you shall reckon all your Azimuths. Count thence to the Limb and back again (if need be) in the said Equinoctial line, till you have made 90 degrees, there is your Meridian, as far distant from the Limb, as the East point was from the Ascendent. Follow this Meridian to the Finitor, and there he shewes you Culmen Caeli, and the Parallel there cutting shewes the Altitude thereof. Now may you find every degree of the Ecliptique above the Ho­rizon, if you know but what Ascends, or Descends, or Culmi­nates; and of every such degree the Parallels shew you the Al­titude, and the Meridians shew his Azimuth, if you begin your numbring from the East or South Azimuth.

Example. When ♋ 24 degrees was Ascending (as in the Example before used) as by consequence ♈ 24. in Nonagisimo gradu, ♂ was in ♉ 4. 45 minutes, and had but 3. or 4. minutes South Latitude: I would know ♂ his Altitude, and Azimuth, setting go the Finitor above the Equinoctial line 42 degrees (which is the Altitude of Nonagesimus gradus) I say, because the Nonagesimus gradus at the end of the Finitor in the Limb, is ♈ 24. therefore I must count back 10. 45 minutes toward the Ascendent for Mars, and there the Parallel 41 degrees with 10 minutes cutteth the Finitor, for the Altitude of ♂, and the 14th. Meridian East-ward from the Limb gives me his Azimuth, which if I begin to reckon from the East point, falleth out to be almost the 40th. Azimuth from the East. Mars his Latitude here is not regarded.

CHAP. LIX. The Nonagesimus gradus, and his Altitude and Azi­muth given, as in the former Chapter. How in the same Projection to get the Altitude and Azimuth of any Planet or Star, by his Longitude and Latitude.

YOur Palnisphear set as in the former Chapter: you shall number the Longitude of the Star upon the Finitor, (here Ecliptique) beginning at the Descendent or Nonagesimus gardus; and in the Azimuth serving his Longitude, count his Lati­tude by the Almicantars, at the end of which account is the Stars place for this time. The Parallel cutting there shewes his Al­titude, and the Meridian cutting there shewes his Azimuth, if you count from the East point as you were taught in the former Chapter.

Example. Lucida Pleiadum was in Longitude ♉ 25. 10 minutes, Latitude 4 degrees 00 minutes North. Therefore from the Nonagesimus gradus ♈ 24. I number in the Finitor toward the Ascendent 31. 10 minutes, and there is the Longitude of Lucida Pleiaedum; in the Azimuth that cuts here I go up North­ward 4 degrees, and there I make a prick for Lucida Pleiadum. Now the Parallel 38 ½ shewes me his Altitude, and the 48th ½. Meridian from the Center, shewes me that Lucida Pleiadum is gone 48 ½ in Azimuth from the Ascendent, but from the East point onely 12 degrees 10 minutes.

CHAP. LX. The Altitude and Azimuth of any Star taken, and ei­ther the Ascendent, Nonagesimus gradus, or Culmen Caeli known: How by the same Nonagesimal Projection to find the Stars Longitude and Latitude.

IF you know either the Ascendent, Nonagesimus gradus, or Culmen Caeli, you have enough to put your Planisphear in the Nonagesimal Projection, by the former Chapters. And your Planisphear so set, you shall seek out the Meridian, [Page 130]which standeth for the Azimuth in which you observe the Star; and therein number from the Equinoctial line the Altitude ob­served: the Azimuth and Almicantar cutting there shew the Longitude and Latitude of the Star inquired. If the Azimuths reach not the place of the Star, turn the Reet half round, and let the Zenith and Nadir points change places; and your turn is served.

Example. Febr. 13, 1657/8. I observed (somewhat near) that ♃ was gone West-ward from the Meridian in Azimuth 14 de­grees, and that his Altitude was 61 degrees; Sirius was then in the Meridian, by which I have the Ascendent, Culmen, and No­nagesimus gradus, any or all of them given. For when in the Equinoctial Projection I bring Sirius to the Meridian line, it is all one as if I had set the Suns place to the hour of the Night (by Chapter 46.) and I see there Culminates with Sirius ♋ 7. 10 minutes, whose Meridian Altitude (by the 46.) is 61. 5 minutes; and I see ♎ 5 ½ ascending in my Horizon, and ♈ 5 ½ descending; therefore ♋ 5 ½ is Nonagesimus gradus, which is 90 degrees di­stant both from the Ascendent and Descendent: his Altitude (by Chapter 55.) 61. 10 minutes almost. Therefore I set the Fini­tor 61. 10 minutes above Meridies, (as Chapter 58.) and in the Finitor at the Limb I count ♋ 5 ½ Nonagesimus gradus; thence I go inwards in the Finitor 1. 40 minutes where I come to ♋ 7. 10. the degree of Culmination; this degree is cut by the 4th. Meridian from the Limb, whereby I learn that this 4th Meridian will be the Meridian of my place, and that the Amplitude of the Nonagesimus gradus, and likewise of the Ascendent, is 4 degrees. Now to place ♃ in the Mater, I count his Azimuth first, begin­ning from the Meridian of my Place now found. First I reckon up from the Culmen Caeli to Nonagesimus gradus in the Limb 4 degrees, (for so much the Nonagesimus gradus is West of the Meridian) and thence back again, I tell to the 10th. Meridian from the Limb, which maketh the 14th. Azimuth from Me­dium Caeli, in which Azimuth I observed Jupiter in that Meri­dian; used here for the 14th. Azimuth) I reckon the Altitude of Jupiter from he Equinoctial line 61 degrees, and at that Alti­tude I make therein a prick for the place of ♃: And imediate­ly I see this prick standeth a quarter of a degree above the Finitor, shewing the Latitude of ♃ 15 minutes North, and it is cut by al­most the 5th. Azimuth from the Limb, which sheweth me that [Page 131]the Longitude of ♃ is 4. 50 minutes less then the Longitude of Nonagesimus gradus; and therefore that ♃ is in ♋ 0. 40 mi­nutes.

CHAP. LXI. The Latitude and Azimuth of a Star, and either the As­cendent, Nonagesimus gradus, or the Culmination given, to find his Longitude.

YOur Planisphear being set in the Nonagesimal Projection, (as in the former Chapter) seek the Meridian that serveth for the Azimuth of the Star, and mark where it cutteth the Almicantar serving for the Parallel of the Stars Latitude. The Azimuth cutting there shewes the Longitude, which you shall reckon from the Ascendent, or Descendent, or Nonagesimus gradus; whose Longitudes are known, as was shewed in the for­mer Chapter.

Example. Suppose ♃ his Azimuth observed 14 degrees from South Westward (as Chapter 60.) and suppose his Latitude known 15 minutes North: (Though I know the Tables make Jupiters Latitude here divers minutes less, that matters not to our purpose here) I say where the 10 ^th Meridian (which by the for­mer Chapter is the 14. Azimuth in this posture of my Plani­sphear) cutteth the Almicantar 0 ¼ serving for Jupiters Latitude, there cutteth an Azimuth which gives me Jupiters Longitude, as in the former Chapter.

CHAP. LXII. To find the Parallactical Angle; that is, what Angle the Azimuth maketh with any point of the Ecliptique, by the Altitude of that point, and of the Nonage­simus gradus.

NUmber on the Label from the Center the Complement of the Altitude of the point proposed, (which may be known by Chapter 56.) and at the end of it make a prick, and having (by Chapter 55. or otherwise) the [Page 132]Altitude of Nonagesimus gradus, turn the prick you made on the Label to touch the Almicantar, which is Complement of that Altitude: then in the Limb of the Reet between the Finitor and the Label is the quantity of the angle.

Example. In Chapter 56. the Altitude of the Sun in the middle of the Eclipse which hapned March 29. 1652. 10. hours 32 minutes a. m. was 41. 47 minutes, and the Altitude of Nona­gesimus gradus 41. 58 minutes; wherefore I make a prick on the edge of the Label at 41 ¾ counted from the Limb, (or I count the Complement hereof from the Center, and make the prick) and having turned that prick to the fourty second Almicantar from the Zenith, I find the Label shewing 87 degrees in the Limb of the Reet, the quantity of the angle.

But because the Label here cutteth the Almicantar so slope that you can hardly observe the just point of Intersection, I will shew you another way.

The Complement of the Altitude of Culmen Caeli, the distance of the point proposed from Culmen, and the Complement of Al­titude of the said point, make a Triangle; whose 3 sides are all known, or may be known by the Chapters foregoing: Therefore by the first Probleme of Obliquangled Triangles you may find the angle.

Example. I set the Complement of the Suns Altitude Z ☉

[Page 133]for A B in the Limb, and marking where the 54 ^th Parallel, counted from the Pole meets with the Almicantar 23. 41 mi­nutes, (counted from the Zenith,) there I find the Azimuth 93. 19 minutes, shewing the greater angle at the Sun, whose Supple­ment 86. 41 minutes is the lesser angle sought; and this lesser angle below the Ecliptique is alwayes Westward in the West Quadrant, and contrary in the East. In the first way you re­solve the Rectangled Triangle Z. 90. ☉, by the Probleme 2. Rectang. and third Variety: in the second way you resolve the Oblique Triangle Z ☉ C by Probleme 1. of Obliqueangled Triangles.

CHAP. LXIII. To find the Parallax of Altitude of the Sun, or Moon.

THe true Altitude of the Sun, or Moon, ought to be ob­served in the Center of the Earth, whereto the Tables are conformed: but because we dwell upon the Super­ficies of the Earth almost 4000 English miles from the Center, therefore the Planets seem lower to us then indeed they be. Suppose a Man observed the apparent Altitude of the Moon at C on the Superficies of the Earth to be E G 10 de­grees; he that could observe it in the Center at B might find the

true Altitude F G almost 11 degrees. Now the smal angle at D (being equal to the difference of these Altitudes, and subtended on [Page 134]this side by the Semidiameter of the Earth B C, and on the fur­ther side by the arch of the difference of these Altitudes in the Primum mobile,) is called the angle of the Parallax of Altitude. This Parallax is greatest in the Horizon, and decreaseth as the Planet riseth higher, till in the Zenith it vanish into nothing: be­cause there the line drawn from the Center falls into the line drawn from the Superficies of the Earth to the Planet, and makes therewith no angle at all.

To get the Parallax for any Altitude proposed, you must first get the Horizontall Parallax out of some Astronomical Tables: for it varyes according to the Planets distance from the Earth; which is not alwayes the same; yet the Suns Horizontal Parallax you may alwayes reckon to be about 3 minutes, and the Moons Horizontal Parallax to be at the least 50 minutes, and at the most 68 minutes. This had, I number the Horizontal Parallax in the Limb from the Equinoctial line, and thereto lay the Label; and number the Altitude of the Planet on the Label from the Limb, and the Parallel that cuts that Altitude shewes the Pa­rallax desired. And note here, that for every minute of the Horizontal Parallax you may reckon 5, 10, or 20, times so many, so that your Label rise not beyond 10 degrees in the Limb; so shall you attain the minutes more exactly.

Example. March 29. 1652. the Altitude of the Sun in the middle of the Eclipse was 41. 47 minutes, and his Horizon­tal Parallax according to Lantsbergius, 2 minutes 18 seconds, for which I number 2 degrees 18 minutes from the Equinoctial line, and thereto set the Label; and so I find the 43 ¾ degrees of the Label to cut the Parallel 1 degree 30 minutes, which I am to accompt 1 minute 30 seconds, the Suns Parallax for this Alti­tude. Likewise the Moons Horizontal Parallax according to Lantsbergius, was then 62 minutes, her Altitude the same with the Suns, or very near; I set the Label therefore to make an an­gle of 6 degrees 12 minutes with the Equinoctial, and so I find the Parallel 4 ⅔ cutting the 41 ¾ degree of the Label; which shewes the Parallax of the ☽ in that Altitude 46 ½, accounting e­very degree 10 minutes, as here I had appointed them to signifie.

CHAP. LXIV. The Parallactique Angle, and the Parallax of Altitude given, to find the Parallax of Longitude and Latitude.

IF the Azimuth or Circle of Altitude make no angle with the Ecliptique, but be co-incident with it (as where the E­cliptique cuts the Zenith) then doth the Parallax of Alti­tude vary the Longitude only; and so much as the Parallax of Altitude is, so much is the apparent Longitude of the Planet greater then the true Longitude in the Eastern Quadrant of the Ecliptique, and so much lesser in the Western.

If the Azimuth make a Right angle with the Ecliptique (which it may do only in Nonagesimo gradu) then doth the Parallax of Altitude vary the Latitude onely; and so much as the Parallax of Altitude is, so much must be added to the apparent North Latitude, or subducted from the apparent South Latitude, to make the true Latitude of the Planet North or South. If the Azimuth cut the Ecliptique with Oblique angles (as most commonly it hapneth to do) then doth the Parallax of Altitude vary both the Longitude and Latitude. And the nearer the Pla­net is to the Nonagesimus gradus, the greater is the Parallax of Latitude, and the Parallax of Longitude less: and contrarily the further the Planet is from Nonagesimus gradus, the greater is the Parallax of Longitude, and the Parallax of Latitude the less.

The Parallax of Altitude is alwayes the Hypotenusa, and the Parallax of Longitude and Latitude are the legs of a smal Rect­angled Spherical Triangle, The Paralla­ctical Triangle. which may be called the Parallactical Triangle, and the leg which hath the Parallax of Longitude is a segment of the Ecliptique, or of a Parallel near it, and the leg which hath the Parallax of Latitude, is a segment of a Circle of Longitude passing through the apparent place of the Planet, and through the Poles of the Ecliptique, and cutting the Ecliptique, or his said Parallel at Right angles, as in the Figure.

• C is the apparent place of the Planet.
• B is his true place, in which he must be seen from the Center.
• A B C is the Parallactique angle.
• B C is the Parallax of Altitude.
• B A the Parallax of Longitude.
• C A the Parallax of Latitude.

[Page 136]

Wherefore by the third Probleme of Rectangled Triangles, Book 3. 5. you may presently get both the legs.

Example. The Parallactical angle B, was found Chapter 62. to be 86. 41 minutes, and the Parallax of the Moons Alti­tude, (Chapter 63.) to be 46 ½ minutes for the same time. Here therefore having laid the Label to 86. 41 minutes from the E­quinoctial, I number in the Label from the Center 46 degrees and an half (in stead of 46 minutes and an half the Parallax of Altitude) and I find that the 46 ⅓ Parallel cutteth the said 46 ½, degree of the Label; by which I know that the Leg C A for the Parallax of Latitude is 46 ⅓ very near, (for the ☽ here being near the Nonagesimus gradus all her Parallax almost goes into Lon­gitude,) but B A of my Triangle is covered by the head of my Label. Nevertheless I may see his measure in any of the Pa­rallels to be 3 ½ minutes for the Parallax of Longitude, for it is the 3 ½ Meridian from the Axis, which cutteth 46 ½ of the Label: and if I had not this shift, I might have my choice of o­ther shifts, shewed Book 3. the 8, 9, 10, 11, 12, and 13 ^th. Chapters.

The Suns Parallax of Altitude Chapter 63 for the same time was found 1 minute ½. therefore laying the Label to the Paral­lactique angle (as before) I number on the Label for the side B C (the Suns Parallax of Altitude, being 1 minute ½) 9 degrees, so every degree here signifieth 10 seconds; and I find there cutting almost the 9 ^th Parallel, shewing me that C A the Parallax of the Suns Latitude is 1 ½ minute almost (that is almost as much as his Parallax of Altitude) and there cutteth also the Meridian 0 ⅔, shewing me that the side B A, Parallax of Longitude, is almost 7 seconds.

[Page 137]The Sun therefore (though he never have Latitude) by rea­son of his Parallax, appeared in the middle of this Eclipse to have South Latitude 1 ½ minute, the Moons true Latitude was then by Lantsbergius his Tables 45 minutes, 24 seconds North; so that by this accompt the Sun and Moons Centers were distant in La­titude 46 minutes, 54 seconds: but when out of this distance you have subtracted 46 ⅓ for the Moons Parallax of Latitude, there remains 34 seconds for the apparent distance of the Centers of the Sun and Moon. But by Observation, I found them distant 1 minute 48 seconds: for the digits Eclipsed at Ecton, were 11. 22 ½ minutes; and so perhaps might I have found by my Planisphear, or some what near, had it been large enough, and had I regarded every minute and second precisely in setting down this Example, which were more then needed for my purpose in this place.

CHAP. LXV. To find the Moons Latitude, by her distance from either of the Nodi, called Caput, and Caudi Draconis.

AS the Ecliptique crosseth the Equator with an angle of 23. 30 minutes for our Age, so the Orbite or Circle in which the Moon moveth crosseth the Ecliptique: but the angle of Inclination is not alwaies nor long the same: for in the Conjunctions of the Sun and Moon the angle is ever 5 00 minutes, and increaseth to the time of the Quadrature, when it is found 5 degrees 16 minutes; thence it decreaseth to the Opposi­tion, where it is again but 5 degrees, as in the Conjunction; thence it increaseth again to 5 degrees 16 minutes in the latter □, and a­gain thence decreaseth to 5 degrees in the ☌.

Get by the Astronomical Tables the quantity of the angle made between the Ecliptique and the Orbite of the Moon, (which in all Conjunctions and Oppositions, and therefore in all Eclipses is 5 degrees, as was now said) and get also by the like Tables the Moons distance from the nearest of the Nodes: then may you find the Moons Latitude, by the Probleme 3. Rectangled Triangles, Book 3. 5. just as you use to find the Suns Declination, by his Longitude and greatest Declination.

Example. The ☽ in the former Case was distant from Caput Draconis by Lantsbergius Tables 8 degrees 43 minutes. I lay [Page 138]the Label from the Equinoctial line to 5 degrees in the Limb, and counting in the Label from the Center 8. 43. I see there the Parallel of 0 ¾ (that is, 45 minutes, or 46 minutes) crossing for the Moons Latitude.

CHAP. LXVI. To find the Dominical Letter, the Prime, Epact, Easter day, and the rest of the moveable Feasts for ever, by the Calender, discribed Book 1. 11.

AN Example, shall serve here instead of a Rule. For the Year 1657. I would know all these: wherefore I seek the Year 1657. in the Table of the Suns Cycle, and over a­gainst it, I find 14. for the Year of the Cycle of the Sun, and D for the Dominical Letter. And note here, that every Leap-year hath 2 Dominical Letters (as 1660. hath A G) and the first ( viz. A) serveth that Year till February 25, and the second (G) for the rest of the Year. And note that these letters go alwayes back­wards when you count forwards (as B A, then G F, &c. not F G, and then A B) as you may see by the Table.

Then in the Table of the Cycle of the Moon, I have for the Year 1657. the Prime 5. the Epact 25. Those had, I go to the Table for Easter, and seek there in the first rank the Prime 5. and under it in the middle rank stands E; that is not my Dominical Letter; therefore I seek not backward, but alwayes forward in the middle rank, till I come to my Dominical Letter D. and under it I find in the third rank March 29. upon which Easter day falls this Year 1657. The rest of the moveable Feasts may be had by their distances from Easter, which are alwayes the same. One­ly for Advent Sunday, remember that the next Sunday after November 26 is Advent Sunday. Read Book 1. 11. and that will sufficiently instruct you with this Example.

CHAP. LXVII. To find the age of the Moon, by the Epact.

REmember first that the Epact begins with March, which must be here accounted the first Moneth: Then if you [Page 139]add to the Epact the number of the Moneth current, and the number of the day of the Moneth current, the sum or the excess above 30, is the Moons age.

Example. January 20. 1656. According to the accompt of the Church of England, (who begin the Year with March 25. which was the Equinoctial day about Christ time) the Epact is 14. January is the 11 ^th Moneth, and the 20 ^th day is proposed; now add 14, 11, and 20. together, they make 45. out of which I take 30. and there remains 15. the Moons age.

This Rule is of good use, not onely to find the age of the Moon, and so her changes to a day, but also for examining of Chronologie, where the time is most certainly reckoned by E­clipses. But you must note, that if you apply this Rule to the Years past before Anno Dom, 1600. then for every 312. Years that the Year proposed precedes Anno Dom. 1600. you must subtract one day out of the age of the Moon, found by this Rule.

Example. [...]icius lib. 1. Reports, That in the beginning of Tiberius Caesar's reigne there was an Eclipse of the Moon, and Temporarius saith, that whereas Augustus died Aug. 29. (I think he should say 19.) this Eclipse hapned Sep. 27. I would know whether it were possible for an Eclipse to happen that day, sup­posing the beginning of Tiberius to be in August, Anno Dom. 14. and Anno Periodi Julianae 4727. The Prime for that Year is 15, and the Epact 15. by Book 1. 11. add now to the Epact, for September 7. and for the day of the Moneth 27. and the sum is 49. out of which subducting 30. I leave the Moons age 19. but because Anno Dom. 14. precedes Anno Dom. 1600. 5. times 312. Years, therefore out of 19 I subduct 5. and there re­maines 14. the age of the Moon, corrected for September 27. Anno Dom. 14. Therefore it was about the full Moon: and it is possible the Moon might be Eclipsed then, as Temporarius saith. But it could not be Eclipsed September 27. Anno Dom. 13. for then the Epact being 4. the age of the Moon by the same Rule was 3. neither could it happen Sep. 27. Anno Dom. 15. for then by the same Rule the age of the Moon, was 25. at what age the Moon was far from her opposition to the Sun: and therefore could not be Eclipsed.

CHAP. LXVIII. To find in what Parallel and Climate a Place is, by the Latitude given.

PArallels in Geography are lesser Circles Parallel to the E­quator, and passing through the Zenith of a Place, and suc­ceeding one another at such distance that at every Pa­rallel the length of the day is varyed a quarter of an hour.

A Climate is such a Parallel as altereth the length of the day half an hour. The Parallels and Climates begin from the E­quator, under which the day is alwayes equal to the Night, and each 12 hours long: hence they count the Parallels and Cli­mates Northward, and Southward: but because the Earth was not so far known to Ptolomy and the Ancient Geographers, as it hath been to those of later Times, therefore there is great difference between the Ancient and later Geographers about the number and quantity of the spaces contained by them: as among others Kerkerman Syst. Geography, lib. 1. hath shewed.

Yet may they easily be found to every Mans mind, by the Planisphear in the Meridional Projection, thus. Find by 4. 17. what is the Semi-diurnal arch of the Sun in ♋, out of which, take 6 hours, and look how many quarters of an hour the double of the residue containeth, so many Geographical Parallels is the place removed from the Equator, and half so many Climates.

Example. I find the Semi-diurnal arch in our Latitude to be 8 hours 16 minutes, in the Tropique of Cancer; out of which ta­king 6. and doubling the residue, I have 4. 33. which is more then 9 half hours, or more then 18. quarters: so much our lon­gest day exceeds 12 hours; therefore we should be past the 18 ^th Parallel, and 9 ^th Climate, viz. in the beginning of the 10 ^th Cli­mate, and 19 ^th Parallel.

CHAP. LXIX. The Longitude and Latitude of two Places given, to find their Distance.

WHat Longitude and Latitude in Geography are, and how they differ from Longitude and Latitude in A­stronomy, hath been shewed, Book 4, 5.11.

If the places differ only in Latitude, and have one Longitude, bearing full North or South one from another; then take their difference of Latitude, by subducting the less out of the greater, if the places have both North Latitude, or both South Latitude: or take the sum of their Latitude, if one be North, and the other South. Then for every degree of this difference or aggregate number, you may reckon 69 ½ English. miles of the Statute, which ordaineth 1760. Yards to be a Mile: but of Eng­lish Miles measured by common estimation, there go not above 60. to a degree; so that every such Mile that you Travel North or South shall alter your Latitude about one minute. If they differ in Longitude onely, and have no Latitude, but be both un­der the Equator, you shall reckon in like manner for every de­gree they differ in Longitude 69 ½ Miles of distance.

In all other Cases you have a Triangle soluble by the second Probleme of Obliquangled Triangles: of which Triangle the Complements of Latitude make the two comprehending sides, and the difference of the Longitudes of the places is the angle comprehended between them, and the third side is the arch of the distance of the places; which when you have found in degrees and minutes of a great Circle, you may turn into Miles as before: mark how the distance of two Stars is found by their Longitude and Latitude given, Chapter 37. in the same manner may you find the distance of two Cities or Towns.

Example. I would know the distance of London from Jeru­salem, The Complement of the Latitude of London is 38. 28 minutes: the Complement of the Latitude of Jerusalem is 58. 05 minutes: the difference of their Longitudes 46. 0 minutes. I set the Zenith to the Latitude of London in the Limb, that is 38. 28 minutes from the Pole, so the Limb is the Circle of Longitude in which London standeth, then I seek the 46 Meridian from [Page 142]that side of the Limb where Zenith is set for London, for that 46 Méridian is the Circle of Jerusalems Longitude: (because the difference of Longitude is 46.) Now because Jerusalems Lati­tude is 31. 55. and the Complement thereof, or distance from the Pole 58. 5 minutes, I walk on in the 46 Meridian till I come where the 58th Parallel from the Pole crosseth him, and there is the place of Jerusalem: the Azimuth that goes hence to the Zenith is the nearest way from Jerusalem to London: what Azi­muth this is I regard not, for I enquire not the angle at London, but I observe by the Parallels how many degrees there be in him between the places of Jerusalem and the Zenith, and I find 38 degrees 20 minutes: which being resolved into Miles is 2300. Miles of common estimation, but Miles of the Statute 2664. the distance of London from Jerusalem.

CHAP. LXX. The Latitude and distance of two Places given, to find the difference of Longitude.

THe Triangle will stand as in the former Chapter: there by two sides and the angle comprehended you sought the third side, by Probleme 2. Obliquangled Triangles: here by three sides given, you seek an angle by Pro­bleme 1 Obliquangled Triangles.

Make the Pole, Pole: and set the Zenith to the Latitude of one of the places, as you did London. (Chapter 69.) 38. 28 mi­nutes from the Pole, then number the Complement of the Lati­tude of the other place from the Pole, by the Parallels, and the distance of the two places from the Zenith by the Almicantars; and where the last Parallel and last Almicantar meet is C of your Triangle: (see Book 3. 14.) Now count how many Meridi­ans there be between C and the Limb, so many degrees is the angle at the Pole sought, for the difference of Longitude.

Example. Having the distance of London from the Pole 38. 28 minutes, and of Jerusalem from the Pole 58. 5 minutes, and the distance of London from Jerusalem 2300. common English Miles, (of which 60. make a degree) I set the Zenith for London 38. 28 minutes from the Pole in the Limb, then because Jeru­salem is distant from the Pole 58. 5 minutes, I go to the 58th [Page 143]Parallel from the Pole, and lay one finger or the point of a bod­kin on him; and because London is distant from Jerusalem 38 degrees 20 minutes, I count from the Zenith to the Almicantar 38. 20 minutes; now where this Almicantar crosseth the Pa­rallel last found, there is C of the Triangle, and the place of Jeru­salem: and you may see that you must cross 46. Meridians before you can go thence to the Zenith in the Limb; which sheweth that the angle at the Pole for the difference of Longitude is 46.

CHAP. LXXI. To find what degree of the Ecliptique Culminates in an­other Country, at any time proposed, if the difference of Longitude be known.

IN the Equinoctial Projection, Bring the Suns place to the hour proposed, by help of the Label: and in the Noon-line you see presently what degree Culminates in your Coun­try. (as Chapter 46, is shewed.) Now to know this for another Town, set the Label so many degrees from the Noon-line as the difference of Longitude requires, and that Eastward, if the place proposed be East, or Westward if it bear West; and so the La­bel shall cut the degree of Culmination, for the place proposed.

Example. If it be demanded what degree is Culminating at Jerusalem March 10. at 10. a clock before noon, I will set the Suns place ♈ 0. to the hour; and I see upon the Noon line, which is our Meridian, there Culminates ♒ 28 almost. Now for the Meridian of Jerusalem I must lay the Label 46 degrees East­ward, that is from Meridies towards Oriens, and look what Star or degree of the Ecliptique is then cut by the Label, that is then Culminating in the Meridian at Jerusalem; (as here I find ♈ 17 ½.) for in this Projection the Label (lay him where you will) is a Meridian.

CHAP. LXXII. To find what a Clock it is in another Country, by know­ing the hour at Home, and the difference of Longitude.

THis is done easily enough without an Instrument: for if you turn the difference of Longitude into hours and minutes, and add the same to your hours for any place which lies Eastward, or subtract the same for any place which lies Westward, you shall make the hour of the place.

Example. The difference of Longitude between London and Jerusalem is 46. or being converted into time 3 hours 4 minutes: therefore adding this to the time at London, I say, when it is noon at London, it is 4 minutes past 3 a clock after noon at Je­rusalem: and when it is 2 a clock at London, it is 5. and 4 mi­nutes at Jerusalem.

If you will do it by the Planisphear, you shall do it in the E­quinoctial Projection, thus. Whereas the Limb of your Rect is graduated into 360 degrees, if you distinguish the hours also at every 15th degree, beginning at the Zenith (which shall be 12) and numbring thence in the Limb of your Reet to your right hand or Westward, 1, 2, 3, &c. then shall you need to do no more but set the Zenith to the difference of Longitude, East or West from your Meridian, as the strange place happeneth to be situate: for then the Label laid to the hour of your Country in the Limb of the Mater, shall shew the hour of the other Coun­try in the Limb of the Reet. And so the Zenith being laid to 60 degrees Westward, which is the Meridian of the Isle of Barba­dos the Label laid to Meridies shall cut in the Limb of the Reet 8 of the clock before noon: which sheweth me that when it is noon with us, it is at Barbados but eight in the morning.

The end of the Fourth Book.

CHAP. LXXIII. The Longitude and Latitude of one Place known, and the Rumb and distance of a second Place, to find both the Longitude and Latitude of the second Place.

SEt the Zenith to the Latitude of the first Place, then seek the Azimuth which serveth for the Rumb of the second Place, and in that Azimuth count his distance from the Zenith: where this distance ends there is the second Place, whole Latitude is shewn you by the Parallel which cut­teth him, and the Meridian cutting there also, shews his Lon­gitude.

Example. Let Z be London, and because Je­rusalem beareth from London al­most S b E or 77 ½ from South Eastward, there­fore I choose the Azimuth 77 ½ Z K. there­in I number Je­rusalems di­stance from London Z I 2300. miles or minutes, that is 38. 20. minutes. Now in the Triangle Z P I, I may find P I the complement of Jerusalems Latitude 58. 05. minutes, and Z P I the difference of Longitude 46, which must be added to the Lon­gitude of London to make the Longitude of Jerusalem.

CHAP. LXXIV. The Latitudes and Distance of two Places given, to find the Rumb, and the difference of Longitude.

COunt in the Meridian from P (the Pole) the comple­ment of the Latitude of the first place, and thereto set Z the Zenith. Count also from P the complement of the Latitude of the second place, and lay your finger on the Parallel at which your number ends. Count also from Z the di­stance of the places in Degrees and Minutes, and note the Almi­cantar at which this number ends: where this Almicantar crosseth the aforesaid Parallel there is C of your Triangle; (but here marked I,) Look what Azimuth cutteth here, it sheweth the Rumb: and the Meridian that cutteth here, (if you count his distance from the Limb) shews the difference of the Longitude of the places. This is so plain from Chapter 69, 70, and 73, that it needeth no example. The same Scheam serveth these 4, Chapters,

CHAP. I. The Preface.
Of the kinds of Dyals.

ALthough Gnomoniques pertain to Astronomy, yet I think it not amiss, for the ease of the Reader in finding them, to place the Gnomonical Problemes in a distinct Book by themselves.

Suns Dyals may be reduced to two sorts.

Some shew the hour by the Altitude of the Sun, as Qua­drants, Rings, Cylinders, &c. for the making whereof you must know the Suns Altitudes for every day, or at least every 10th day of the year, and for every hour of those dayes: which Altitudes you may find immediately upon this Planisphear, as in a Table made to your hand, for any Latitude, by Book 3.25. and so make them of any shape according to your mind.

The other sort shew the hour by the shadow of a Gnomon, or Style, Parallel to the Axis of the World: and of those I treat cheifly in this Book. Those be all Projections of the Sphear upon a plain which lies Parallel to some Horizon or other in the World. And if upon such a plain the Meridians onely be projected, they shall suffice to shew the hour, without projecting the other Circles, as the Ecliptique, the Equator with his Pa­rallels [Page 146]of Declination, the Horizon with his Almicantars and Azimuths, which are sometimes drawn upon Dyals, more for or­nament, then for-necessity.

CHAP. II. Theorems premised.

FOr the better understanding of the reason of Dyals, these Theorems would be known.

• 1. That every plain whereupon any Dyal is drawn, is part of the plain of great Circle of the Heaven: which Circle is an Horizon to some Country or other: that the Center of the Dyal represents the Center of the Earth and World, and the Gnomon which casteth the shade representeth the Axis, and ought to point directly to the two Poles. And if upon the Center of the Dyal you fasten a Label with Sights of equal Al­titude, and keeping your eye in the line of the Sights turn this La­bel round, you shall thereby describe in the Heavens that great Circle wherein your Dyal-plain lies, and see where it cuts our Ho­rizon, and how much it is Elevated above it on one side, and de­pressed on the other.
• 2. That those Dyal-plains Geometrically are not in the very plains of great Circles; for then they should have their Centers in the Center of the Earth, from which they are removed almost 4000. Miles: and in truth they lie in the plains of Circles Parallel to the said Horizons, but so near them, that Optically they seem to be the plains of those Horizons: because the Semidiameter of the Earth beareth so smal proportion to the Suns distance, that the whole Earth may be taken for one point or Center, without any perceivable error.
• 3. That (as all great Circles of the Sphear, so) every Dyal-plain hath his Axis, which is a straight line passing through the Center of the plain, and making right angles with it: and at the ends of the Axis be the two Poles of the plain, whereof that a­bove our Horizon is called the Pole Zenith, and the other the Pole Nadir of the Dyal.
• 4. That every Dyal-plain hath two faces or sides: and look what respect or situation the North Pole of the World hath to the one side, the same hath the South Pole to the other, and [Page 147]these two sides will alwayes receive 24 hours; so that what one side wanteth, the other side shall have; and the one is described in all things as the other.
• 5. That (as Horizons, so) Dyal-plains are with respect to the Equator divided into 1. Parallel or Equinoctial, 2. Right, 3. Oblique.
• 6. A Parallel or Polar Dyal-plain maketh no angles with the Equator: but lies in the plain of it, or Parallel to it. Such Dials are Scioterica Orthognomonica, that is, have the Gnomon erected on the plain at Right angles, as the Axis of the World is upon the plain of the Equator: because the Axis and Poles of the Dyal be here all one with the Axis and Poles of the World: and the hour lines here meet all at the Center, making equal angles, and dividing the Dyal Circle into 24. equal parts as the Meridi­ans do the Equator, in whose plain the Dyal lies.
• 7. A right Horizon or Dyal-plain cutteth the Equator at right angles, and so cutteth through both the Poles of the World. Therefore such Dyals are Paralielognomonical: that is have the Gnomon Parallel to the plain, and so the hour lines and the hour lines all Parallel one to another: because their plains though in­finitely extended will never cut the Axis of the World. Yet have those Dyals a Center, (though not for the meeting of the hour lines) viz. through which the Axis of the Dyal Circle passeth, cutting the plain at right angles, and cutting also (near e­nough for the projecting of a Dyal) the Center of the World.
• 8. An Oblique Horizon or Dyal-plain cutteth the Equator at Oblique angles: such Dyals are Scalenognomonical: that is, have for their Gnomon the side of a Triangle whose angles vary accord­ing to the more or less Obliquity of the said Horizon: and the Gnomon shall alwayes make an angle with the plain of so many degrees as the Axis of the World maketh with the plain; or as either of the Poles of the World is Elevated above the plain.
• 9. Every Oblique Horizon is divided by the Meridians or Hour Circles of the Sphear into 24. unequal parts: which parts, are alwayes lesser as they are scarer to the Meridian of that Ho­rizon or plain, and greater as they are further off: and on both sides the Meridian of the plain the hour Circles which are equal­ly distant in Time, are also equally distant in Space: whence it is that the divisions of one Quadrant of your Dyal plain being known, the division of the whole Circle is likewise known.
• [Page 148]10. The Hour-lines in an Oblique Dyal are the Sections of the plains of the Hour-circles of the Sphear with the Dial plain. And because the plains of great Circles do alwayes cut one ano­ther

  [diagram]

  in halfs by Diameters, which are straight lines passing through the common Center, therefore lines drawn from the Center of the Dyal to the Intersections of the Hour-circles with the great Circle of the plain, shall be those very Sections, and the very Hour-lines of the Dyal.
• 11. Every Dyal-plain (being an Horizon to some place in the Earth, as was said Theorem 1.) hath his proper Meridian; which is the Meridian cutting through the [...]oles of the plain, and ma­king Right angles with the plain. If the Poles of the Dyal-plain lie in the Meridian of our place; then is the Meridian of the plain all one with the Meridian of the place; and the Gnomon or Style shall stand erected upon the Noon-line, or line of 12 a clock, as in all direct Dyals: but if the plain decline, then shall the sub­stylar or line wherein the Gnomon standeth, which is the Meridi­an of the plain, vary from the Noon-line, which is the Meridian of the place: and this variation shall be East if the Declination [Page 149]of the plain be West, and contrarily: because the visual lines by which the Sphear is projected on Dyal-plains, do all like the beams of a Burning-glass intersect or cross one another in a cer­tain point of the Gnomon (to be assigned at pleasure, and called Nodus) and so do all place and depaint themselves on the Dyal-plain beyond the Nodus the contrary way.
• 12. Dyals are most aptly denominated from that part of the Sphear where their Poles lie: though some Authors have chosen to denominate them from the Circles in which their plains lie: as the Dyal-plain which lieth in the Equinoctial or Parallel to it, is called by many an Equinoctial plain: but I concur with those who would rather call it a Polar-plain; because the Poles thereof are in the Poles of the World.

CHAP. III. How to draw an Horizontal or Vertical line, upon any plain.

BEcause in the Delineation of most Dyals the Horizontal or the Vertical line of the plain must first be drawn, before you can place the hour-lines; I will shew you first how to draw either of them. Know this first that they cross one ano­ther at Right angles in the Center of the Dyal: and therefore if you can draw either of them, you may draw the other also. Also in upright plains as are Walls a Quadrant or a Square with a Plumbet applied to the Wall, will shew you how to draw both of them very easily. Or if you hang a Plumb-line quiet before the Wall when the Sun shineth, the shadow of it shall be a Ver­tical line at any time.

But if the plain incline or recline, you shall set to it a Square with a Plumbet, and thereby first draw the Horizontal line; for when the Plumbet (which must play in an hole) hangs Parallel to one side of the Square, a line drawn by the other shall be Ho­rizontal: that drawn, you shall lay your Square flat on the plain, and draw the Vertical from any point of the Horizontal at Right angles by your Square

Another way is this. Hang your Planisphear by the handle with his Plumbline, so that the Plumbline fall upon one of the Diameters; then setting your Label and Sights to the other Dia­meter, [Page 150]look through the Sights, and mark where the visuall line cuts the plain neer to one side, and there make a prick, then di­rect your Sights to the other side of the plain, and make another prick, (your Label and Plumbline being still at Right angles as before) by these two pricks draw a line, and it shall be an Hori­zontal line. And note, that all lines Parallel to an Horizontal line be Horizontal: and all lines Parallel to a Vertical line be also Vertical.

CHAP. IV. How to make the Polar Dyal, and how to place it.

THe plain of the Polar Dyal lieth in the Equinoctial: where the 12 cheife Meridians or Hour-circles divide both the Equinoctial and this plain into 24 equal parts. The Gnomon stands upon the Center at Right angles with the plain. You may learn to make him only by the 6. Theorem of the second Chapter.

Take your Planisphear in the Equinoctial Projection, and there is your Dyal ready made on the Limb, and the Hours al­ready marked. Erect now a wyer or thred Perpendicular upon the Center, or hold a Square to the Center, so that his top be e­qually distant from all the parts of the Circle: and there is your Gnomon placed.

To place this Dyal do thus. Having (by Book 4.3.) found a Meridian line, if you cross it with another line, that shall be an East or West line. Have also in a readiness a square board up­on which you may fasten your Planisphear or Dyal-plate with pins, screws, or wax, so that the Noon-line may be Parallel to two sides of the board, and the East-line to the other two: then set the North side of the board in the East line, even now found, and raise the South side by a Quadrant to the height of the E­quinoctial; and so may you place your Dyal in any Window if it be made upon a loose round plate; but if the plate be square, you need not the board to place it.

Note that both faces of this Dyal must be divided, and the Gnomon must appear on both sides, like the stick in a Purre (or whirligig) which Children use: otherwise you must turn him upside down, as oft as the Sun passeth the Equinoctial.

CHAP. V. How to make the South Equinoctial Dyal, or Pa­rallelognomonical Dyal direct.

THe Equinoctial Dyal we call that which hath his Poles in the Equinoctial Circle: of which there be three kinds.

• 1. The direct or South Equinoctial Dyal, which fa­ceth the Meridian directly, not looking from him to one side more then to the other, having his Poles in the Intersections of the E­quinoctial and Meridian.
• 2. The East or West Equinoctial Dyal, which may also be called the Equinoctial Horizontal Dyal: for an Horizontal Dyal declining just 90 degrees from the South or North, becomes an Equinoctial Dyal, as well as Horizontal: because there his Poles light upon the Intersection of the Horizon with the Equinoctial. And though this Dyal be of kin to both, yet his Gnomon shews that he should be sorted rather with the Equinoctial Dyals then with the Horizontal: for he is Parallelognomonical: these two sorts be regular, having their Poles in the four notablest points of the Equator: the third is somewhat Irregular, but may be brought to Rule.
• 3. The Equinoctial Dyal declining, whose Poles happen any­where else in the Equator between the Horizon and Meridian.

Now to make the first of these, the South Equinoctial Dyal, draw first an Horizontal-line upon the plain, and cross it with a Vertical-line, by Chapter 3. The Intersection of these is the Cen­ter of your Dyal, and would be chosen about the middle of the plain. Now your Planisphear being fastned to a square board, as in the former Chapter, you shall set the North side of the said board in the Horizontal-line of the plain, so that the Axis or East line of your Planisphear may be Parallel thereto, and the Noon-line (or Equator line of the Mater) may point directly to the Center, making Right angles with the plain, or at least with the Horizontal line thereof: (for it is not material whether the board be upright, slope, or lie flat upon the Dial plain) then placing your Sights first in the Noon line, they shall point to the Center for the point of 12. thence remove your Label to 1. of the [Page 152]

[Page 153]clock in the Limb, and they shall point out in the Horizontal line of the plain, the point of Intersection for one a clock, where you shall make a prick: In like manner remove your Label to 1, 2, 3, 4, and 5. in order, making pricks in like manner▪ When you remove your Label to the Axtree line for 6 you shall find that the line of the Sights maketh no Intersection with the plain, but runneth Parallel to it; because the Sun is then in the Horizon of this Dyal, where he projects the shadows of all upright things infinite. And as you found the points for the hours afternoon, so may you by like reason find the points for the 5. morning hours, and their quarters also, if you please: which had, if by these points found you draw lines Parallel to the Vertical line of the plain (which is here the Meridian of the plain, and of the place) they shall be the true hour-lines. And the Gnomons edge must stand over the Meridian, and Parallel to it, at the same di­stance that the Axtree line of your Planisphear was situate in pro­jecting the hour points. If you cannot fix your Planisphear on a board, as abovesaid, or if your plain require a Gnomon of a greater or lesser height, you may upon any board presently draw so much of your Planisphear as serves for this purpose. Or to say more breifly, Do but make the height of your Gnomon Ra­dius, and the Tangents of 15, 30, 45, 60, 75. shall give the distances of the hour points in the Horizontal line, on both sides, from the Center of the Dyal, as may appear in the Figure.

CHAP. VI. How to make the East Equinoctial Dyal, or the West.

THis plain is a Right Horizon of those People who dwell under the Equator, distant from us 90 degrees of Lon­gitude; as the South Equinoctial-plain of the last Chap­ter was the Horizon of those who dwel under the Equator, in the same Longitude with us. Therefore these Dy­als are in all points alike: onely the Substylar-line which in the South Equinoctial Dyal is 12. in this East Equinoctial Dyal is but 6. in the Morning for our Country, because of the difference of Longitude.

To protract this on the wall or plain, first draw a Perpendi­cular or Vertical line by Chapter 3. as A B. Pitch one foot of [Page 154]your Compasses in any convenient point thereof (as C) and with the other foot draw a blind arch from some lower point of the Perpendicular South-ward (as D E.) In this arch number the Elevation of the South Pole above this Perpendicular (that is, the Complement of your Latitude, being with us 37.45 minutes) and from that degree to the Center of the said arch draw the line E C, which is the very Axis of the World, pointing to both the

Poles; Cross this Axis at Right angles with the line C F, and that shall be a Diameter of the Equator and the Contingent-line, as they call it. Now choose a fit point in this Contingent for the Substyle or 6. a clock point, as at G, and thereto by the square board mentioned in the former Chapter, set Oriens, or Occident, of your Planisphear; and so the Label set to the several hours in [Page 155]the Limb of the Planisphear shall shew the hour-points in the Contingent line: onely for 12. the line of the Sights will not in­tersect the plain at all, and therefore here is no Noon-line, as there was no 6. a clock-line in the South Equinoctial Dyal but so soon as the Sun leaving the plain makes the shadow of the Gno­mon infinite, then it is Noon.

Lastly, draw a line Parallel to the Contingent line at such di­stance as the plain will afford, as the line E I, and to this you shall protract your hour lines, drawing them from every point of the Contingent to this; so that they make Right angles with the Con­tingent and with this Parallel, even as the rounds of a Ladder do with the sides, but that the distance of the rounds of a Ladder are equal, and these distances be unequal. The Gnomon must be set like a Bridge Perpendicularly over the 6. a clock hour-line, the edge that casteth shadow being Parallel to it, and of such height as the line K G of your Planisphear, or so that if the Gnomon fall, his edge may lie in the line of 3. or of 9. of the clock. This also may be made speedily by help of the Tan­gents, as the South Equinoctial Dyal.

For the West Equinoctial Dyal, it is made like the East in all points; onely it shews but the after Noon hours, as the East shews but those of the fore noon. When you have drawn on paper the East Dyal, and set it by guess in its Situation, go on the West side of it, and you may see through the paper the picture of the West Dyal, and so will the back side of the West Dyal shew you the true picture of the East.

CHAP. VII. How to make the Declining Equinoctial Dyal.

ANy Declining plain may be so Reclined that he shall be­come a Right Horizon or Equinoctial plain, and at what Reclination this shall happen you may easily find by Chapter 19.

When you are sure that your plain is a Right Horizon, having his Poles in the Equinoctial, you shall by the same 19 ^th Chapter get the Oblique Ascension of the Noon-line, and the difference of Longitude; which had, the Dyal shall be made in this man­nar. See the second of the five Scheams of Declining-Recliners, [Page 156]Chapter 20. Draw upon your plain (by Chapter 3.) an Hori­zontal line D d (answering to D d in the said Scheam) therein take a point at pleasure, as the point Z, (answering to Z in the said Scheam) and thereupon draw above the Horizontal line West­ward, if the Declination be West (as the said Scheam shall direct you) the Quadrant of the plain D P T; (answering D Pro T of the said Scheam) herein number from D the arch D P, which is the Oblique Ascension of the Noon line, and draw Z P for the Axis and common Section of all the Meridians. It appeareth now

[Page 157]that no Dyal can be drawn upon the very plain of a Meridian, for there all the hour lines will be represented by one line, which is the Axis of the World; therefore you must draw such Dyals on a Parallel plain, as you did the two former after this manner.

Set the line M E N for the Axis of the World or Gnomon, and prop him up over the line P Z with two props of equal height, and Perpendicular to the plain, and make the point E (which standeth Perpendicularly over Z) the Center of the World; then from this Axis or Gnomon mounted in the Air, shall the hour-lines be projected distinctly, and all of them shall be Parallel to the Axis and one to another, as it hapneth in all sorts of Equinoctial Dyals. The line Z P shall remain now onely the Meridian of the plain or Substylar. And to find the hour-lines you shall do thus.

Draw through Z an Equinoctial or Contingent line E Z, ma­king Right angles with the Axis M E N or Z P, then setting Z O equal to Z E, draw upon the Center O (with an extension of the Compasses,) the arch of the Equinoctial b Z d, or the Pa­rallel arch passing by D. Then number in this arch from the Substyle (Westward if your plain decline West, or Eastward if it decline East) the difference of Longitude, and where it ends there is the point of Noon in this arch: from that point begin to divide the said arch by fifteens of degrees, or 24 ^th parts of the whole Circle. And remember, that when you come to 90 de­grees from the Substyle, you need divide no further, for the Sun is no longer upon this plain. Also you may leave out those hours at which the Sun is alwayes under our Horizon, as the hours from 8. at Night, to 4 in the Morning: then lay a Ruler from the Center O to every one of these divisions of the Circle, and where the Ruler cuts the Contingent, there make points for the hours respectively, and through these points you may draw the hour-lines Parallel to the Substyle, of what length you please; and mark them from the Noon line Eastward 1, 2, 3. &c. because the Suns Diurnal course is Westward, and the course of the sha­dow is contrary.

He that will may make use of his Planisphear for dividing the hours, as was taught Chapter 4, and 5. or use a Quadrant, or a Scale of Chords, or the Tables of Tangents with a Sector, or a Scale of equal parts. But it needs not.

Note that this Dyal may compare with the hardest: however [Page 158]M ^r Blagrave and other Dyalists have omitted it, as seeming easy: and here Wittekindus, (to whom all later Dyalists are much be­holden) and after him Fale were mistaken, using the Declinati­on of the plain, where they should have used the difference of Longitude in the making of this Dyal.

CHAP. VIII. Of the kinds of Oblique Dyals.

WHat an Oblique Dyal is, and why it is so called, hath been shewed Chapter 2. They be

• Regular.
• Irregular.

The Regular lie in some notable Circle of the Sphear; as, 1. The Vertical Dyal, whose plain lieth in the Hori­zon: for which cause many call it the Horizontal Dyal. 2. The South and North Horizontal Dyal, whose plain lies in the East Azimuth: and it is commonly called the South or North Erect Direct Dyal. As for the East and West Dyals, they be­long to another place, as was said Chapter 5.

The Irregular are such as lie Oblique to the Horizon, as Re­clining or Inclining Dyals; or lie Oblique to the Meridian, as Decliners: or else Oblique to both, as Recliners or Incliners Declining; which are esteemed the hardest of all, because of their double Irregularity, though by the Planisphear they are made almost as easily as the rest.

The Declination of a plain is the Azimuthal distance of his Poles from the Meridian of the Place, East or West.

The Reclination is the distance of his Poles from the Zenith and Nadir of your Place. Inclination is the nearest distance of the Poles of the plain from your Horizon. And whatsoever the Reclination of the upper face of a plain is, the Inclination of the lower face is the Complement thereof.

CHAP. IX. How to make the Vertical Dyal.

IN the Meridional Projection the Finitor being set to the La­titude of your Place, you shall see the Limb which is your [Page 159]Meridian, and the Axtree-line which is the sixt hour-circle, divi­ding the Finitor into 4 Quadrants; and the rest of the Meridi­ans dividing every Quadrant alike. Mark now at what de­gree numbred from the Limb, every hour-circle (that is, every 15 ^th Meridian being a ragged or blacker line) cutteth the Finitor; at the same distance shall the same hour-circle cut the Limb of your Dyal in the plain.

Example. To

make a Vertical Dyal for our Lati­tude 52 degrees 15 minutes. I set the Finitor to this Lati­tude, and because I see the Meridian and Axtree divide the Finitor into 4 e­qual Quadrants, therefore I open my Compass to the Se­midiameter of my Planisphear, or of a­ny other Circle which I have at hand already divided, and with that measure I draw a Circle, and cross it square with two Diameters dividing it into four equal parts: of which Diameters I appoint one to be the Noon-line, and at the North end thereof I write 12. the other Diameter then is the East line, and at both ends thereof I write 6. Then I count in my Planisphear how many degrees from the Meridian or Noon line every other Circle cutteth, and I find that the first cutteth 11.58 minutes, wherefore I take with my Compasses from off the Limb of my Planisphear, or o­ther Circle used for this purpose, the space of 11.58 minutes, and set it in the Circle of my Dyal from 12. both wayes, for the hours of 1. and 11. and because I find the second hour Circle cutting the Finitor at 24. 32 minutes, therefore I take in like manner the Chord of so many degrees from the Limb of my Planisphear, and set it also in the Circle of my Dyal both wayes from 12. for the hours of 2. and 10. And when you have done the like for the 3, 4, and 5. hours, you shll draw lines from [Page 160]those points to the Center, and set to those lines from the North Eastward 1, 2, 3, 4, 5. and Westward 11, 10, 9, 8, 7. And because the Sun in Summer is above our Horizon more then 2 hours before and after 6. (for I see the 7 ^th and 8 ^th hour Circles intersecting the Tropique above the Finitor) and because hour lines equally distant from the Meridian or Axtree cut like de­grees of the Horizon (as my Planisphear here shews me) and so shall make equa langles at the Center; therefore laying my Ru­ler to 7 in the Morning I prolong that line beyond the Center to the other side of the Circle for 7 at Night: likewise by the prolongation of 8 in the Morning, I make 8 at Night, and by the prolongation of 4 and 5 of the after Noon, I make 4 and 5 of the Morning hours.

Lastly, for the Gnomon, set your Compasses to the Chord of the arch of the Poles Elevation in the Limb: that is, measure in the Limb from the Pole to the Finitor, and setting that di­stance in the Circle of your Dyal from 12. either way, make a point, through which if you draw a deleble line from the Center, you have between this line and the line of 12. the angle of your Gnomon, by which when you have shaped him, you must set him upright over the 12 a clock line, with the point of the said angle at the Center, and all is done.

CHAP. X. How to make the South and North Horizontal Dyal.

THis is usually called the Erect Direct Dyal, and belongs to an upright Wall looking full North or South: and the plain of it lies in the East Azimuth, which on the Planisphear in the Meridional Projection is represented by the Axis of the Reet.

The Finitor set to the Latitude, as in the former Chapter, mark where the hour Circles cut the Axis of the Reet, which is the proper Horizon of this Dyal; you shall find the first cutteth 9. 20 minutes from the Meridian, the second 19. 30 minutes, the third 31. 30 minutes, the fourth 46. 45 minutes, the fifth 66. 24 minutes; the sixt 90. And you shall see the North Pole depressed under this plain, as much as is the Complement of our Latitude, and the South Pole as much Elevated above it.

[Page 161]1. Wherefore for the South Dyal, draw an Horizontal line about the top of your Dyal plain, which shall be the hour of Sixes, from the midst whereof let fall a Perpendicular, which shall be both the Vertical and the Meridian, both of the Place, and of the Plain, wherein the Gnomon must stand Elevated 37. 45. mi­nutes or the Complement of your Latitude toward the South Pole.

From the Center (which is the point from whence the Per­pendicular falls,) draw a Semi-circle beneath the Horizontal line, of equal Semi-diameter with that of your Planisphear or of any other Circle which you have divided; and in this Semi-circle set off on both sides from the Perpendicular or Meridian line the distances of the hours before found, making pricks in the Semi­circle, and thereto drawing lines from the Center, and setting figures, after the same manner as you did in the

former Chapter; and your Dyal is done. This Dy­al shews the Hours from 6 in the Morning to 6 at Night; the o­ther Hours before and after 6 be­long to the North face of this Dyal.

Another way. Because the Al­micantars may oft obscure the Intersections of the Hour Circles with the Axis, you may avoid that inconvemence, if you reduce this Dyal to a Vertical Dyal.

For the South Horizontal Dyal being the very Vertical Dyal of those People that live 90 degrees Southward from us, that is, in South Latitude 37. 45 minutes, if you set the Finitor to the Latitude 37. 45 minutes, you shall see the sections of the Hour Circles with the Finitor more [...]pparently, and thereby make your Dyal.

[Page 162]2. For the North face, Imagine you had for you. Gnomon a wyre thrust aslope through the Center of the plain from the South side Northward and you will presently conceive that in the North Dyal the Horizontal or 6 a clock line will be lowest, and that the Gnomon will turn upwards toward the North Pole as much as he turned downwards on the other side: and that all the hours save 4, 5, and 6. in the Morning, and 6, 7, and 8. at Night may be left out in our Latitude; because the Sun shineth no longer upon it: and those hour-distances you may find, and set off from the 12 a clock line, or from the 6 a clock line, as you did the hours of like distance in the South face.

Another general and pleasant way to delineate the opposite face of any Dyal, see hereafter in the end of the 12 ^th Chapter.

CHAP. XI. How to Observe the Declination of any Declining Plain.

ALL Perpendicular plains, as Walls, lie in the plains of one of the Azimuths: which plains alwayes cut both Zenith and Nadir, and the Center of the Earth. As in the figure. Z is Zenith and Nadir: E S W N Ho­rizon, E W is the Base or ground-line, or any

Horizontal line drawn upon a Wall or plain looking full South or North; his Poles are at S and N in the Me­ridian, wherefore he declineth not, but lieth in the East Azimuth E W.

A B is a Wall or plain declining East by the arch S p, to which E B or W A are e­qual, for so much as the Wall bendeth from the East Azimuth, so much doth his Pole at p decline or bend from the Meridian.

[Page 163]1. Now to find how much any plain declineth, and so in what Azimuth he lies, one good way is this: when the Sun begins to inlighten the Wall, or when he leaves it, then is the Sun in the same Azimuth with the Wall; take at that instant his Altitude, and thereby get his Azimuth (according to Book 4.27.) and that is the Azimuth of the Wall.

2. Another way, First draw upon the Wall an Horizontal line, by Chapter 3. then your Planisphear being fastned to a Square board (as in Chapter 4.) set one side of the board to that Horizontal line or Parallel to it and fix there your board and Planisphear level, by the help of a Square set under him like a bracket, the place your Label and Sights in one of the Diame­ters of your Planisphear, and mark when the Sun comes into the line of the Label, casting the shadow of one Sight upon the o­ther, if the Label be then in the Diameter which is Parallel to the Wall, then is the Sun at that time in the Azimuth of the Wall: if the Label be in the other Diameter which is Perpendicular to the Wall, then the Sun coming to it is in the Azimuth of the Pole of the plain.

Now having the hour, or the Altitude of the Sun get his A­zimuth (by 4.27,) the same is the Azimuth of the Wall or plain, if the Label were Parallel to the Wall; or the same is the Azimuth of the Pole of the plain (that is the very Declination) if the Label stood Perpendicular to the Wall.

3. Another way, If you have not time to watch till the Sun come into the Azimuth of the Wall or the Vertical of it, which cutteth the Pole thereof, then get the Suns Azimuth by the said Book 4.27, when you can, and at the same time Ob­serve by your Label the Suns Horizontal distance from the Pole of the plain, and by comparing these together you may easily gather the Declination of the Wall: as in Example.

I observed the Sun to be gone West from the Pole of the plain 70 degrees, and by the Altitude of the Sun then taken, I found his Azimuth 60 degrees: here I reason thus, The Sun is gone from the Pole and Vertical of the Wall 70 degrees, and from the Meridian but 60 degrees, therefore the Meridian lies between the Pole of the plain and the Sun and because ☉ p is 70. and ☉ S 60. therefore S p the Declination of the plain, is 10 degrees the difference of 70. and 60, and the Declination is East, for the Sun is neerer to the Meridian then to the Vertical of [Page 164]the plain: and thus if you draw a rude Scheam of your Case, you may soon reason out the Declination, better then do it blind-fold by the rules commonly given.

And by these two last wayes you may take the Declination, not only of upright plains, but of Recliners also, for which the first way will not serve,

CHAP. XII. How to make a Declining Horizontal Dyal.

HEre three things are required. For besides the distances of the several hours from 12, and the Elevation of the Gnomon, which are requisite to the making of all Di­rect and Regular Dyals, we must here know also the Declination of the Gnomon, which some call the distance of the Substyle from the Meridian, or the distance of the Meridian of the plain from

the Meridian of the Place. For in all Dy­als the Noon­line in the Me­ridian of your Place, project­ed on the Dy­al, and in all Horizontal or Mural Dyals, not Reclining or Inclining, the Noon-line is a Perpendi­cular cutting the Center of the Dyal, how much soever they De­cline.

But Declining Dyals which look awry from our Meridian, have a Meridian of their own, which is called the Meridian of the plain, and the Substyle (because the Style or Gnomor stands upon it) and is indeed the Meridian of that Place, where this De­clining Dyal would be a Vertical Dyal, and where the Substyle [Page 165]would be the Noon line: and to this Substyle the hours of the plain are alwayes so conformed, that the nearer they be to the Substyle the narrower are the hour spaces, and contrarily: be­cause the Meridians do so cut every oblique Horizon, that is, thickest near the Meridian of the place: and this Declining Dyal (being a stranger with us) followeth the fashion of his own Coun­try, and so hath his narrowest hour spaces near his own Meridi­an, rather then ours Now as that is the Meridian of our place which cutteth our Horizon at Right angles, passing through his Poles Zenith and Nadir, so the Meridian of any plain is that which cutteth the plain at Right angles, and passeth through his Poles.

You may find all these requisites in the Meridional Projection, not only for one, but for all Declinations, lying as in a Table be­fore you, with admirable ease and delight; for there is no Decli­ning Wall or Horizontal plain, but we have an Azimuth in the Reet which shall picture him: and look how the Meridians di­vide these Azimuths, so do they divide the Horizons or Circles of the Declining plains. The Pole of any Azimuth is found in the Finitor 90 degrees distant from him; the Meridian that cuts the Pole of the Azimuth, cuts also the Azimuth, and the plain thereby represented at Right angles, and is the Meridian of the plain or Substylar, (Chapter 2. Theorem 11.) and the de­grees of that Meridian between the plain and the next Pole of the World are the Elevation of the Pole above the plain: and so the Elevation of the Gnomon or Style, and the arch of the plain com­prehended between this Meridian of the plain and the Limb, is the Declination of the Gnomon, or distance of the Substyle from the Meridian, or distance of the Meridian of the plain from the Meridian of the Place. What would you more?

Example. If a Wall Decline East 30 degrees. I say, because the face of the Wall looketh 30 degrees from the South East­ward, therefore the plain, which lieth 90 degrees from his Pole, is in the 30 ^th Azimuth from the East Northward: therefore I go to the 30 ^th Azimuth from the East line of Axis, counted ci­ther way, and take that Azimuth and his Match (which is e­qually distant from the Axis) for the very picture of my Decli­ning plain.

Then seeking the Substyle or Meridian of the plain. I say the Pole of the plain is in the Finitor at the 30 ^th Azimuth from Me­ridies [Page 166]in the Limb, (because the plain it self is the 30 ^th Azimuth beyond the Axis) the Meridian that cuts this Pole is the 36 ¼ (exactly 36. 8 minutes, the number whereof shewes me the difference of Longitude between our Country and the Country of this Dyal. This 36 ¼ Meridian, being the Meridian of the plain, I follow toward the Pole, and find him cutting both the arches of my plain on both sides the Axis: but I regard the cut­tng only in that arch which is nearest to the Pole, because there the angle looks more like a Right angle, and there is the nearest distance of the Pole from the plain, and there I see the hour spaces least: from that Intersection therefore, I reckon in the same Meridian to the Pole 32 degrees and perhaps a minute more, (you may find it by Calculation,) this is the Elevation of the Pole above the plain, and of the Gnomon likewise: also from the same Inter­section I reckon in the plain to the Limb or Meridian 21. de­grees 10. minutes, the distance of the Meridian of the plain from the Meridian of the Place: the same is the Declination of the Gnomon, or of his Substyle.

Then for the hours, I begin at the Zenith of the Reet, where is our Meridian, and numbring first toward the Substyle, I seek at what number of degrees from the Zenith the hour Circles cut my plain, and I find as followeth,

Then in the other arch of the plain I have the afternoon hours, thus.

And further I cannot go, because I see the next hour is a­bove 90 from the Substyle; therefore my Dyal receives him not on this side, but on the North side there is use of him.

Now to draw the Dyal, I consider that because the plain de­clines East therefore the Gnomon shall decline West: for the Dy­al being such a projection of the Sphear wherein all the Vi [...]ual lines cross in the Nodus of the Gnomon, and thence disperse them­selves again toward the plain, therefore that which is East in the Sphear will be expressed West on the plain, and contrarily, (as [Page 167]was shewed Chapter 2. Theorem 11.) Also I consider that how­soever the plain be turned East or West, the Gnomon's place is fixed, because it is a part of the Axis of the World, or a line Pa­rallel to it. Now therefore if I turn a South Dyal, and make him Decline East, and hold the Gnomon unmoveable; the West side of the Dyal will approach nearer to the Gnomon, as reason and sence will tell me: likewise the hours which are found on the same side of the Meridian or Noon-line with the Substile must be set the same way with it from the Noon-line in the Dy­al. Therefore having drawn an Horizontal line E W on the Wall, from the Center taken at A, I let fall the Perpendicular A B for the Noon-line, then upon the Center A. I draw a blind Se­mi-circle with the Semi-diameter of my Planisphear, or of some Quadrant, as E B W and therein I prick down the Substyle and the hours, after the manner used the 10 ^th Chapter.

And if you would draw the North Dyal of this plain, do but prolong those

hour lines, and the Sub­style upwards beyond the Center, and you have the North Dyal, above the Ho­rizontal line E W, as the South Dyal below it; and note, that be­cause in our Latitude the Sun sets soon after 8 in Summer, therefore the 3 hours next be­fore and after mid-night may be left out in this Dyal, and in all others which must serve in our Latitude.

This is the most ready way to delineate the opposite face of a­ny Dyal. See another way to make this Declining Horizontal Dyal, Chapter 21.

CHAP. XIII. How to Observe the Reclination or Inclination or any Plain.

WHat Reclination and Inclination are, hath been shew­ed Chapter 8.

All Reclining and Inclining plains have their Ba­ses or Horizontal Diameters lying in the Horiz ontal Diameter of some Azimuth: but the top or Nonagesimus gra­dus of the plain from the Horizon leaneth back from the Zenith of your Place, in the Vertical of the plain (which is the Azi­muth cutting the plain at Right angles) so much as the Reclina­tion hapneth to be; and the Pole of the plain, on that side the plain inclines to, is sunk as much below the Horizon as the No­nagesimus gradus of the plain is sunk below the Zenith, and the opposite Pole is mounted as much.

Let E S W N be

Horizon, Z the Ze­nith. E W the Hori­zontal Diameter of the plain and of the East Azimuth E A W a plain not Declining but Reclining South­ward from the Zenith by the arch Z A 50 degrees and his oppo­site face Inclining to the Horizon according to the arch A S 40 de­grees, the Pole of the Reclining face is at P in the Meridian (which here is also Verti­cal of the plain) and is Elevated 50 degrees in the arch N P e­qual to the arch of Reclination Z A: and the Pole of the Incli­ning face is depressed as much on the other side under the Ho­rizon.

To find the quantity of the Reclination you shall draw a Verti­cal li [...]e on the plain, by Chapter 3. and thereto apply a long Ruler [Page 169]which may over-shoot the plain either above or below: to that Ruler apply any Semidiameter of your Planisphear, or of any Qua­drant; and the degrees between that Semidiameter and the Plumb line shall be the degrees of Reclination.

Or stick up in the Vertical line two pins of equal height, and Perpendicular, and placing your self either above or below the plain, as you find most easy, direct the Sights of your Planisphear or Quadrant to the heads of the two pins being in a right line with your ey; and the Plumbet shall shew the Reclination on one side the Quadrant, and the Inclination (which is alwayes Complement thereof) on the other.

CHAP. XIV. How to make a South and North Reclining Dyal.

THe Base or Horizontal line of such a Dyal lieth in the East Azimuth, and his Pole in the Meridian; as you may see in the plain of the former Chapter. In the Meridional Projection having set the Finitor to the La­titude, count from the Zenith the degrees of Reclination North­ward or South-ward as you observed it to be, and remove the Zenith so many degrees the same way, then shall you see pre­sently which Pole is Elevated above the Zenith line, (for that is the picture of your plain) and how much: to which Elevation you shall make your Dyal by the tenth Chapter, remembring to turn the Gnomon upwards or downwards as the North or South Pole is Elevated above the face of your plain.

Example. The plain of the former Chapter was a North plain Reclining Southward 50 degrees, that is, almost to the E­quinoctial: when the Finitor is at our Latitude the Zenith is distant from the North Pole the Complement thereof 37.45. toward the South. Now I must put the Zenith yet 50 degrees more Southward, because my plain Reclines so much that way, and I see that then the North Pole is Elevated 87. 45. minutes, and I see upon the Zenith line or Axis of the Reet how the hour Circles cut my plain almost in equal spaces: if this plain had Reclined but 2. 15 minutes further, he had fallen into the plain of the Equinoctial, and so the Dyal would have been a Polar Dyal, and all the hours would have had equal space, and the [Page 170] Gnomon would have stood Perpendicular, which are the proper­ties of a Polar Dyal, as hath been shewed Chapter 4.

For the opposite face of this Dyal, the general rule given Chapter 12. may suffice.

CHAP. XV. How to make an East or West Reclining Dyal.

AS it hath been shewed Chapter 14, that the base or Hori­zontal line of a South Recliner lieth alwayes in the East Azimuth, so the base of an East Recliner lieth alwayes in the Meridian of the Place. And as all Declining plains lie in some Azimuth, and cross one another in the Zenith and Nadir, by Chapter 12. so these Reclining plains lie in some Circle of Position, and cross one another in the North and South points of the Horizon: which being considered, those East Recliners shall be made as easily as the Decliners Chapter 12.

For these East Recliners be in very deed South Decliners to those that live 90 degrees from us Northward or Southward; and have one of the Poles Elevated as much as the Comple­ment of our Latitude: for the Perpendicular or Plumb-line of those People is Parallel to the Horizontal Diameter of our Me­ridian.

In the Meridional Projection, set the Zenith line to the Lati­tude, and then are the Azimuths Circles of Position, and are also those very East or West Reclining plains, and the Zenith line is the base or Horizontal line to them all, and to the Meridian like­wise; Take any of these Azimuths, and see how the Meridians of the Mater divide him, so shall the Dyal-plain represented by it be divided also. The working is very like that of Chapter 12. Compare the one with the other where you doubt.

Example. I have an East plain Reclining 45 degrees, to which I would make a Dyal. I set the Zenith line to the Latitude 52. 15 minutes and going to the arch of the 45 Azimuth on one side the Zenith line, and his match so many degrees distant on the other side, I take that Circle for my plain, his Center is the Center of the Planisphear: the Meridian or Noon line of the Place in these Dyals is evermore the Axis or Zenith line of the Reet, for he is drawn to the Intersection of the Meridian of the [Page 171]Place (here the Limb) with the plain. The Zenith line here lies Horizontal, therefore the Noon line in these East Recliners must be evermore the Horizontal line of the Dyal, as in all Decliners (Chapter 12.) the Noon line is evermore the Vertical or Perpen­dicular line.

The arch of my plain which is nearest to the North Pole hath his Pole in the Finitor 45 degrees from the Center Southward, and there this Pole is cut by the Meridian 31 ¼. from the Axis, this is the Meridian of my plain, and he is distant from the Me­ridian of the Place 58 ¾. (which is the difference of Longitude) this Meridian I follow to the arch of the plain which is nearest to the North Pole, and so going on in him to the Pole I number in him between the plain and the Pole 34 degrees, which is the Elevation of the Pole, and therefore also of the Gnomon above this plain: and between this Meridian of the plain and the Meri­dian of the Place at Zenith I reckon in the plain 42 ½ for the Declination of the Gnomon.

Wherefore having drawn an Horizontal line N S for the Noon line, I appoint the Center at S the South end, because

[Page 172]the North Pole is Elevated, and drawing a blind arch, I set there­in from the Horizontal line upwards 42 ½. and there draw the Substyle S V.

Then I seek my hour distances, and I find the first hour-circle from the Meridian toward the Substyle cuts in the plain 14 ½, therefore taking 14 ½ in the blind arch from the Noon line to­ward the Substyle, I set 11. for so it is, and not 1. as you may persently perceive, if you hold but a Book or a Trencher after the Situation of your plain, somewhat near by guess, and consider which way the shadow must move, reason will tell you it moves downward in this Dyal from 11. to 12. &c. then I see the second hour-circle cuts the plain at 25 ½ for 10. the next at 34 ⅔. for 9. the next at 43. for 8. which happneth a little above the Substyle, as he ought, for the difference of Longitude is almost 60. viz 58 ¾. as before, next 51 ½. for 7. next 61 ⅓ for 6. next 73 ⅓ for 5. next 88 ½ for 4. and further I need not go in our Country.

Then in the other arch of the plain I find 19 ½ for 1.45 for 2. 71 for 3. and these I put in their places, as in the Figure. The Gnomon must stand square upon the Substyle, at an angle of 34 degrees.

Note that the Reclination must alwayes be reckoned from the Limb inwards upon the Finitor, because where the Finitor tou­ches the Limb there is our Zenith for this turn. Inclination is reckoned from the Zenith line, which here is both the Diame­ter of the Horizon, and Horizon itself.

For the Opposite or Inclining face of this Dyal-plain, use the general way I shewed you, Chapter 12. that is, strike the Sub­style and all the hour lines through the Center: and set the same figures to every hour line beyond the Center which he had on this side, and set the Gnomon upon the Substyle downward to behold the South Pole, and it is done. And so by the Incli­ning Dyal, if you had him first drawn, you might presently make the Recliner.

CHAP. XVI. How to find the Arches and Angles that are requisite for the making of the Reclining Declining Dyal.

BEfore you can Intelligently make a Reclining-Declining Dyal, which is the most Irregular of all, having two A­nomalies, viz. Declination and Reclination, you must be acquainted with 3 Triangles in the Sphear, wherein certain arches and angles lie which are neefull to be known.

I advise you therefore first to draw (though it be but by aime) an Horizontal Projection of the Sphear, such as here I have drawn for a South Dyal Declining West 50 degrees, and Recli­ning 60 degrees in the Latitude 52. 10. minutes, which shall be our Example.

The Circle E S W N is our Horizon N S our Meridian.

[Page 174]D T d the plain, Z T the Reclination thereof.

D d the Base or Horizontal line of the plain.

V u the Vertical of the plain cutting it right in T, and cutting the Pole thereof at H: for u is the Pole of a plain erected upon D d; but the Pole of the Reclined plain D T d is H.

S u or N V the Declination of the plain.

M P H m the Meridian of the plain, cutting the North Pole at P, the plain in Right angles at R, and the Pole thereof at H.

Now see your three Triangles all adjoyning in this Scheam, viz. D N O and O R P Rectangled at N and R, and P Z H. Obtuse-angled at Z.

It is true that the last Triangle alone may do your work, or the two first may do it without the last: but you shall do well to be acquainted with them all.

In the first Triangle D N O you have given D N 40 degrees the Complement of the plains Declination, N the Right angle of our Meridian with our Horizon, D the Complement of Recli­nation, whereby you may find D O the Oblique Ascension of our Meridian; that is, how many degrees of the plain the noon­line shall lie above the Horizontal line: also you may find N O the Perpendicular Altitude of the noon-line, or the Inclination of the noon-line of the Dyal to the Horizon; (where you shall note that when this Altitude of the noon-line N O is equal to N P the Elevation of the Pole, then is the second Triangle P R O quite lost in the point P, and the plain then becometh a Decli­ning Equinoctial plain) also you may find the angle O called the Position-Reclination for a reason hereafter to be shewed; Wit­tekindus calls it, Complementum repetendum, because he means to have a Bout with it again, to find other arches by it.

In the second Triangle O R P you have given O, as before, for this angle was in the former Triangle, or his equal, (for An­guli precrucem oppositi sunt aequales) R the Right angle of the plain with his Meridian: O P the position Latitude, that is, the Latitude of that Place wherein the Reclining plain O R T d Q, shall be a Circle of Position: this is given if you subtract N O the Perpendicular Altitude of the noon-line, out of N P your La­titude, (this O P is Wittekindus his Differentia Retenta.) And hence may be found O R the Declination of the Gnomon, or distance of the Meridian of the plain from the Meridian of the Place: R P the Elevation of the Pole above the plain in the [Page 175]Plains own Meridian: P the angle between the Meridian of the plain and the Meridian of the Place: this angle is called the difference of Longitude, because it shewes how far the places are distant from us in Longitude, wherein this Dyal shall be a Direct Dyal, without Declination, having his Gnomon in the noon-line of the Place, and and shews also how many degrees of the E­quinoctial, or how many hours and minutes there are between our Meridian and the Meridian of the plain, as the arch O R shews how many degrees of the plain come between the said Meridians. Let this be well observed by Learners.

In the Third Triangle P Z H, you have given P Z the Comple­ment of your Latitude, Z H the Complement of the plains Reclina­tion, and Z the Supplement of the plains Declination.

Hence may be found H P whose Complement is P R the E­levation of the Pole above the plain, P the difference of Longi­tude, H whose measure is R T the arch of the plain between the Meridian of the plain or Substyle and the Vertical line of the plain, the Complement whereof is R D the Substyle's distance from the Horizontal line of the plain.

Every arch and angle therefore in these Triangles is given, or may presently be found by the Problemes of Spherical Triangles, Book 3.

But I shall shew you a short and pleasant way to find them, by setting the whole Scheam at once on your Planisphear, where you shall have them almost all at one view.

For this purpose I use my Planisphear in the Horizontal Pro­jection. But note, that I make use onely of the Reet and Label, and one of the Meridians of the Mater.

Thus I set D d on the Axis of the Reet, D T d on the 60 ^th Azimuth from the Center, V u on the Finitor, Z N on the Label, fixing it in the Limb of the Reet 40 degrees from the Zenith, according to the arch D N. Then in the Label I make a prick with ink at P for the North Pole 52. 15. minutes within the Limb, and another prick in the Finitor at H for the Pole of the plain 30 degrees from the Center; which done, and keeping my Label fixed to my Reet, I turn the Reet till I see some one Meri­dian cutting both the pricks P and H, (as the 15 Meridian from the Axis shall do in this Example) and that Meridian shall serve for the Meridian of the plain for this time.

And by this time I see my three Triangles on my Planisphear, [Page 176]their sides divided into degrees, as a Carpenters Rule into Inches. and I find for the Latitude 52. 10. minutes, that D O the Oblique Ascension of the Meridian is 44.06.

N O the Altitude of the Meridian 20. 22 minutes.

O P the Position Latitude 31. 48. minutes.

O R the Declination of the Gnomon (13. 21. minutes) from the Meridian.

R P the Elevation of the Gnomon 29. 08. minutes, and P H Complement thereof.

H the distance of the Meridian of the plain and Vertical there­of, you may see by the arch of the plain which measureth the angle H, 32. 32. minutes.

Now have you nothing to ask of the third Book, but the angles O and P, and there you have divers wayes to find that.

O the Position-Reclination is 67. 29. minutes.

P the difference of Longitude 26, 00 ½. minutes.

CHAP. XVII. How to find the Horary distances of a Reclining Decli­ning Dyal.

TAke the easiest way first. You have seen Chapter 15. how easily East and West Reclining Dyals are to be made by the Planisphear, because they fall out to be Circles of Position, and are plainly pictured by the A­zimuths.

Now I will shew you how all Reclining Dyals may be redu­ced to East or West Recliners, for some Latitude or other, and so the hour distances found by the Method of the 15 ^th Chapter.

The Circles of Position, as hath been shewed, do all cross one another in the North and South points of the Meridian. Now therefore by the point O where the plain cuts our Meridian, draw a new Horizon O B Q C, and then shall you see your plain in that Horizon to be a very Circle of Position. But now we are gotten into a new Latitude, O P called (before in Chapter 16) the Position Latitude, and we have here a new Reclination, for whereas this plains Reclination in our Latitude is Z D T 60 degrees, his Position Reclination is O, viz. Z O T, or P O R 67. 29. minutes.

[Page 177]In the making of this Dyal therefore, you shall forget your own Latitude, and the plains Reclination in your Horizon, and with this new Latitude and Reclination make the Dyal after the manner of the East Recliner Chapter 15. not regarding the De­clination at all: For the Base of this plain is now fallen into the Horizontal line of the Meridian, and his Declination being just a Quadrant, he becomes a Regular Plain, and neither his Declina­tion nor his Reclination shall now much trouble you.

How to place your Noon-line from the Horizontal or Verti­cal line of the plain, you have found already, and at what di­stances every Hour shall stand from the Noon-line in the plain you shall thus find.

Set the Zenith line of the Reet to your new Latitude P O 31. 48 minutes, and find for your plain the Azimuth 67 ½ from the Axis, because P O R is 67. 29 minutes, as before: seek his Pole in the Finitor, which will be the 90 ^th degree from the said Azimuth or plain, and you shall find that Pole cut by the 26. [Page 178]Meridian: this therefore is the Meridian of my plain, and shall make the Sub-style on the Dyal: his distance from the Meridian of the Place in the Equinoctial is 26. and so much is the angle O P R the difference of Longitude, as before: then follow this Me­ridian of your plain to that arch of the plain which is next the Pole of the World, and you shall number from the plain to the Pole in the said Meridian 29. 08 minutes for the Elevation of the Gnomon P R, as before: and from this Meridian to the Meridian of the Place at the Zenith you shall number in the plain 13. 21 minutes for O R, the Declination of the Gnomon, as be­fore.

Now the crossing of the plain with the Meridian of the Place is the Noon-point in the plain, and that in this case is alwayes in the Zenith, and I see the rest of the Meridians cutting the plain for the Morning Hours, thus.

And for the Evening Hours in the other arch, thus.

And because 7 in the Morning will be shewn by this Dyal in the Summer; to find the distance thereof from the Noon-line in the plain, I set Nadir in the place where Zenith was before, and so I see the 7 ^th hour-circle cutting the plain at 98. 25. minutes. But this is more then I need to do: for having once found 12. Hour-spaces in any Dyal, I can make any of the rest by striking the Hour-line of the same denomination through the Center. As for Example. If I prolong the 7 ^th of the afternoon Hour-lines be­yond the Center, I there make the line of 7a clock in the Morn­ing.

CHAP. XVIII. How to draw the Reclining Declining Dyal.

FIrst draw an Horizontal line, as A B and upon the Center A describe a blind Circle equal to the Limb of your la­nisphear, or of some other Circle which is divided to your hand, that by help thereof you may presently divide this blind Circle into any parts required Then set in this blind Cir­cle the arch D O the Oblique Ascension of the Noon-line 44. 06 minutes from B upwards at C and from C yet upwards (as the Scheam shews you) set the arch O R 13. 21. for the Decli­nation of the Gnomon: and draw the lines A C for the Noon-line, and A D for the Sub-style.

Then set off the distances of the several Hours from the Noon-line

[Page 180]on both sides, viz. the Morning Hours below the Noon-line Westward, and the Evening Hours above it Eastward: as you may be taught by Chapter 2. Theorem 11. and by your own reason.

And lastly set the Gnomon Perpendicular upon the Sub-style A D, making his angle at A equal to the arch P R, found above to be 29. 08. minutes, and your Dyal is done.

CHAP. XIX. How to know at what Reclination any Declination Plain shall become a Declining Equinoctial Dyal-Plain, to be delineated after Chapter 7. And how to find the Oblique Ascension of his Meridian or Substyle, and the difference of Longitude, which are requisite for his Delineation.

THere is no Declining plain but at some certain Reclinati­on cutteth through the Poles of the World and so be­cometh a Right Horizon. Therefore to find whither a Declining Reclining plain do happen to be a Decli­ning Equinoctial plain, you shall observe what the Elevation of the Noon-line N O is; for if that be equal to N P the Latitude, then doth the plain cut the Poles, otherwise not: And at what Reclination any Decliner shall cut the Poles, and so have the Altitude of his Noon line equal to the Latitude, you shall thus find.

Use the Reet and Label in the Horizontal Projection as you did Chapter 16. that is, set the Horizontal line of the plain D d on the Axis of the Reet, then number from the Zenith in the Limb of the Reet the Complement of the Declination, and thereto lay the Label, and having made a prick with ink in the Label for the Pole (52 ¼ from the Limb) mark which of the A­zimuths cutteth that Pole for he sheweth you at what Reclina­tion that Decliner shall cut the Pole and fall into the plain of one of the Maridians. And now you shall have but one Trian­gle to resolve, viz D N P▪ (for the whole Triangle P R O of Chapter 16. is swallowed up in the point of the Pole P) and and D N P hath all his sides known, and the angle D at first [Page 181]Sight; and for P you may find him if you turn the Triangle as Book 3. hath been shewed.

Example. I would make an Equinoctial Dyal in the West Declination 50. degrees, Lay the Label therefore 50. degrees from the Finitor, or 40. from the Zenith, and so the Axis of the Reet represents the plain Declining 50. degrees and the Label represents the North part of the Meridian, and now I see the Azimuth 26 ½ from the Axis cutting the Label in the place of the Pole; therefore I say, that Azimuth represents the Equinocti­al plain which belongs to this Declination.

And now I see the Triangle D N P on my Planisphear, N P on the Label is the Latitude, and also the Altitude of the Noon line 52. 10. minutes, D N Complement of the Declination 40. D P the Oblique Ascension of the Noon line 61. 59. minutes, N is a Right angle, D is Complement of the Reclination 63. 28. minutes, (whose measure in the Scheam is T V) P the Comple­ment of the difference of Longitude, for the difference of Longi­tude it self in the Scheam is H P Z, and the Complement thereof Z P T to which D N P is equal, by the Structure for it is An­gulus pre decussim oppositus: and by any of the four first Problemes of Rectangled Triangles you may find it to be 46. 44. minutes, whose Complement is 43. 16. minutes, the difference of Longi­tude. The Oblique Ascension of the Noon line, and the diffe­rence of Longitude thus found, you shall have enough to make the Dyal by Chapter 7.

CHAP. XX. An Admonition concerning the five several Cases of De­clining Recliners.

BEcause by the diversity of Declination and Reclination, the figure and situation of the three Triangles mentioned Chapter 16. is so charged that you cannot alwayes find them on the sudden, unless you have a firm comprehen­sion of the Sphear in your head and in the Case of the last Chap­ter the middle Triangle is quite lost, having all his sides and ang­les contracted into the very point of the Pole; therefore I have thought good to set down the 5 several Cases of these Dyals in so many several Scheams, and in every Scheam to mark the Tri­angles [Page 182]with the same Letters, that what Case soever shall happen to be proposed; you may have a Scheam ready to direct you.

And to know which Scheam shall serve to express the situati­on of your plain, take these Rules

• 1. If the plain Recline North below the Pole, so that the arch N O the Perpendicular Altitude of the Noon line be less then N P the Elevation of the North Pole, then the first Scheam serves your Case.
• 2. If the plain Recline to the Pole making N P and N O e­qual, you shall use the Second cheam.
• 3. If the plain Recline not so far as the Pole, but make N O greater then N P, you shall use the Third Scheam.
• 4. If the plain Recline Southward, then instead of the Tri­angle D N O you shall use the opposite Triangle d S O where if S O be greater then the Elevation of the Equator or equal to it, you shall use the 4 ^th Scheam. And if it be less, you shall use the Fifth.

And note that in the Fifth Case you may best do you work by the Triangle P H Z alone, (the Triangle P R O being here too big) setting off your Sub-style from the Vertical by the measure of the angle H, or of the arch T R, the Noon line from the Sub­style.

CHAP. XXI. How to make the Declining Horizontal Dyal, another way then was shewed Chapter 12.

THough you have in the former Chapters a perfect Me­thod for the making of all sorts of Dyals which give the Hour by the shadow of the Axis of the World, or a Gnomon set Parallel to it: yet I think it both pleasant and profitable for the Reader to see some other ways whereby the same things may be performed.

For the Declining Horizontal Dyal you shall first find the E­levation and Declination of the Gnomon after this manner. Take your Planisphear in the Horizontal Projection, and for the help of your fancy▪ lay it flat upon a Table, setting the Meridian, that is, the Equator line of the Mater in the Meridian of your Place by aim. Then set the Fini [...]or for your plain; and his Pole [Page 183](which is the Zenith of the Reet) shall be Eastward or West­ward from the Meridian of the Mater so much as you observed the Declination: then seek the place of the Pole in the Meridian, viz. the North Pole 37. 45. minutes Northward from the Cen­ter, if the Wall look North, or the South Pole so much South­ward, if the Wall look South: and it where not amiss if the Poles

were there marked with an Asterisk, or some such note, 38. or 40. degrees from the Center, somewhat near your Latitude, so these Asterisks (if they be not exactly the Poles for your Lati­tude) shall direct you to find the Poles presently, being very near them: as if the Asterisk be at 50. I know 2 ¼ more inwards is my Pole for Latitude 52 ¼. Next look what Azimuth cutteth this Pole, for he shall represent the Meridian of the plain; follow him to the Finitor, and you may number as you go the Elevati­on of the Pole above the plain; and the degrees of the Finitor between this Azimuth and the Center are the degrees of the Declination of the Gnomon or distance of the Sub-style from the Meridian.

2. Having found these two, You shall set your Planisphear in the first Mode of the Meridional Projection: that is, Set the Fini­tor to the Elevation of the Pole above the plain, so shall you have all your Hour distances distinguished upon the Finitor by the Meridians. But here you must carry this in your head, that here the Limb is not the Meridian of your Place, but of the [Page 184]plain; and then to find the Meridian of your Place, number from the Limb in the Finitor the Declination of the Gnomon, and the Meridian cutting there is the Meridian of your place, and stands for Noon: therefore every 15 ^th Meridian numbred from hence (and not from the Limb in this Case) is an Hour line for your Dyal: and look at what distance from the Limb they cut the Finitor, at the same distance from the Sub-style shall the Hour­lines be set in your Dyal plain.

Example. I have a South plain Erect Declining 30. degrees Eastward, (as in Chapter 12.)

And for this, First I set the Zenith 30 degrees from Meridi­es, toward Oriens, and so doth the Finitor represent my plain, and I see the Azimuth 21.10. minutes cutting the South Pole of the Horizontal Projection about 38 degrees from the Center; from the Pole to the Finitor I number in this Azimuth 32.01. mi­nutes, the Elevation of the Pole above the plain; and from the In­tersection of this Azimuth with the Finitor to the Center where the Meridian of the Place meets, I number in the Finitor 21.10. minutes, so much is the Declination of the Gnomon or of his Sub­style from the Noon line.

Secondly, setting the Finitor to the Plain's Latitude 32.01. minutes, I number from the South point of the Finitor inwards 21.10. minutes, and there cuts the Meridian 36.08. minutes from the Limb, shewing me the difference of Longitude, or Equi­ncctial distance of the Meridian of the Place from the Meridian of the plain; for this 36 Meridian is the Meridian of my Place, and therefore I mark him well, he is the Vertical of my Dyal, and also the Noon line. And here I consider that the Sub-style will be Westward from him upon the plain, because it Declines East­ward (by Chapter 2. Theorem 11.) Therefore beginning at Noon, where the 36 Meridian cutteth the Finitor I go 15. to­ward the Limb, and light upon the 21 ^th Meridian from the Sub­style for 11 a clock, and he cutteth the Finitor at 11.35. mi­nutes from the Limb: so the 10 ^th Hour is the sixt Meridian on this side the Limb, and cutteth the Finitor 3.16. minutes: the 9 ^th Hour is the ninth Meridian beyond the Limb, as you come back again, and cutteth the Finitor at 4.44. minutes from the Limb or Sub-style on the other side (that is, Westward of the Sub-style in the Dyal.) In like manner you may gather the distances of the other Hour lines from the Sub-style into a Table, and thereby plot them down as in the Figure.

CHAP. XXII. To make the Reclining Declining Dyal, another way.

HAving found the arches and angles requisite by Chap­ter 16. and platted down your Horizontal and Verti­cal lines, and placed the Noon line above o [...] below the Horizontal line, according as the arch of his Oblique Ascension or Descension requireth, and having placed also the Sub-style in his due situation as is above taught, you may easily find the distances of the several Hours from the Sub-style, as you found them in the former Chapter for the Declining Horizon­tal Dyal.

For when you have set the Finitor to the Latitude of your plain, as there you did, the Limb is Sub-stylar, and if you number thence in the Finitor the Declination of the Gnomon, there shall meet you the Meridian of the Place. Here you shall begin, and take every 15 ^th Meridian forwards and backwards for an Hour line, and observing how many degrees are in the Finitor between the Limb and every one of these Hour lines, so many degrees shall you place that Hour line from the Sub-style in the plain. If you understand the former Chapter this will need no Example.

CHAP. XXIII. To draw the proper Hours of any Declining Dyal.

EVery Declining plain, whether it Recline or not, hath two great Meridians much spoken of. 1. The Meridian of the plain, which is the proper Meridian of that Country to whose Horizon the plain heth Parallel. 2. The Me­ridian of the Place, which is the Meridian of your Country, in which you set up this Declining plain to shew the Hours; and so either of these Meridians Dyals may be conformed. How to draw the Hours of our Country on such a plain is the harder work, because the plain is Irregular to our Horiz on: yet I suppose I have made the way very easy in the former Chapters. But to draw the Hours of the Country to which the plain belongs, is most easy. For if you take the Sub-stylar for the Noon-line, [Page 186]and the Elevation of the Pole above the plain for the Latitude, you may make this Dyal in all points like the Vertical Dyal, after the precept of the 9 ^th Chapter.

CHAP. XXIV. To know in what Country any Declining Dyal shall serve for a Vertical Dyal.

IF the Dyal Decline East, add the difference of Longitude (found as above Chapter 21.) to the Longitude of your Place, and the sum or the excess above 360 is the number of the Longitude sought. If the Dyal Decline West subtract the said difference of Longitude out of the Longitude of your Place, and the difference is the Longitude inquired: but when the Lon­gitude of your Place happens to be less then the difference of Longitude you must add to it 360. before you subtract the differ­ence of Longitude. The Elevation of the Pole above the plain is the Latitude of the Place inquired.

Example. The Declining plain of Chapter 12. will be a Ver­tical plain in the Longitude 61. degrees, and North Latitude 32. degrees, that is, in the Mediterranean Sea between Alexandria and the Isle of Creet. And the Declining Reclining plain of Chapter 16, 17, 18. is Parallel to the Horizon of those that sail in Longitude 359. degrees, and North Latitude 29. degrees that is as Terrestrial Globes and Mapps shew me, between the Azo­res and Hesperides.

CHAP. XXV. To set a Plain Parallel to the Horizon of any Country pro­posed.

IF you can get the Declination and Reclination of such a plain, you have enough to place him in his true Situation. And those may be found by the difference of Longitude and the Latitude of the strange Country, (which are in this Pro­bleme supposed to be given) even as in Chapter 16. you found both those by the Declination and Reclination given.

Example. I would set a plain Parallel to the Horizon of Jeru­salem, [Page 187]to shew me what time the Sun Rises and Sets there any day of the Year, and what Hour passeth at Jerusalem at any time of our day. First I seek by Geographical Tables or Mapps the Longitude and Latitude of Jerusalem, and I find that Jerusalem is removed Eastward from London in Longitude 47 degrees, and that the Latitude there is 32 degrees. or thereabouts. Therefore in the Rectangled Triangle P R O of Chapter 16. I have the angle P 47 degrees difference of Longitude, also the side P R the Latitude of Jerusalem 32 degrees, and hence by the 4 ^th Pro­bleme of Rectangled Triangles Book 3.6. I get P O 42.30 mi­nutes, and by consequence O N 9.45. minutes (because P N is our Latitude) and I get also the angle O 51.40 minutes. And these had, I get by the same Probleme in the adjoyning Trian­gle O N D, both D N 12.05. degrees, the Complement of the Declination inquired, and the angle D 39.23. Complement of the Reclination inquired. Wherefore I conclude that a plain which shall represent here the Horizonof Jerusalem must Decline Eastward 77.55. minutes, and Recline Northward 50.37. mi­nutes. Draw upon this plain the proper Hours of Jerusalem, by Chapter 23. and know that when the Sun leaveth this plain cea­sing to inlighten the upper part of it, then he setteth at Jerusalem, and look how many Hours and minutes the Sun setteth after noon in any Country, so many Hours and minutes he rose before noon.

CHAP. XXVI. How other Circles of the Sphear besides the Meridians may be Projected upon Dyals.

THe Projection of some other Circles of the Sphear be­side the Meridians though it be not necessary for find­ing the Hours yet may be both an ornament to Dyals, and usefull also for finding the Meridian, and placing the Dyal in his due Situation, if it be made upon a moveable Bo­dy, as shall be shewed Chapter 33.

The Circles fittest to be projected in all Dyals for those pur­poses are the Equator with the Tropiques and other his Parallels; which may be accounted Parallels of Declination, as they pass through equal degrees, as every 5 ^th or 10 ^th of Declination: or Parallels of the Signes, as they pass through such degrees of De­clination [Page 188]as the Sun Declineth, when he entreth into any Signe, or any notable degree thereof; or Parallels of the length of the day, as they pass through such degrees of Declination wherein the Sun increaseth or decreaseth the length of the day by Hours or half-Hours.

Also the Horizon with his Azimuths and Almicantars are an ornament to Horizontal and Vertical Dyals; and are likewise use full for projecting the Equator and his Parallels in all Dyals. My purpose is to be breif in this Treatise of the Tumiture here following because I hasten to an end. I shall therefore think it sufficient if I shew you one way to furnish any Dyal with the Circles of the Sphear. Leaving you to devise others which I could have shewn.

CHAP. XXVII. How to describe on any Dyal the proper Azimuths and Al­micantars of the Plain.

FRom any point of the Gnomon (taken at pleasure) let fall a Perpendicular upon the Sub-style; that Perpendicular shall be part of the Axis of the plain, and shall be reputed Radius to the Horizon of your plain. The top of this Radius in the Gno­mon is called Nodus, because you must there set a Knot Bead, or Button to give shade, or else cut there a notch in the Gnomon, or cut off the Gnomon in the Place of Nodus, that the end may give the shadow for those lineaments. Let not your Nodus stand too high above the plain, for then the shadow will fall beside your plain for too great a part of the plains day nor let it stand too low, for then the lineaments will run too close together. A mean must be chosen.

At the foot of this Radius take your Center, and describe a Circle on the plain and divide it into equal degrees; and from the Center draw lines through those degrees infinitely, that is, so far as your Dyal-plain will bear; these lines shall be the Azi­muths of the Horizon of the plain, and shall be numbred from his Meridian or Sub style.

Divide any of these Azimuth lines into degrees, by Tangents agreeable to the said Radius, and having made a prick at every degree, through every of these pricks, you shall draw Parallel [Page 189]

[Page 190]Circles which shall be Almicantars or Parallels of Altitude, to be numbred inwards, so that at the Center be 90. for the Zenith, and from the Center outwards you shall number 80, 70, 60. &c. till you come within 10. or 5 degrees of the Horizon; for the plain is too narrow to receive his own Horizon, or the Parallels near, if the Nodus have any Competent Altitude.

And to divide the said Azimuth lines you use the Tables of Tangents with a Scale of equal parts, or else plot the Tangents thus on paper, set A B equal to the Radius of your Horizon, and with that Radius draw the Quadrant A B C, or A b c, and divide the Quadrant, numbring the degrees from C to B, and ha­ving drawn the Tangent B D, or B d, Parallel to A C, draw lines from the Center through the several degrees to the said Tangent-line, so shall this Tangent-line be divided for your pur­pose: and from it you may transfer the divisions to your plain.

Now if your plain lie in the Horizon of your place, (as the Ver­tical plain doth) these Azimuths and Almicantars may be of some use to shew you the Altitude and Azimuth of the Sun for any time. See them in the Scheam Chapter 30.

But if your plain lie not in the Horizon of your place, then you shall draw the said Almicantars or Azimuths, or so many of them as you shall need, in deleble lines, because here they serve only the Horizon of the plain: yet shall they help you to describe the Equator and his Parallels, with the Horizon of your Place in any Dyal: and when they have done this, unless your Dyal be Vertical, they may be gone.

CHAP. XXVIII. How by help of the proper Azimuths and Almicantars of the Plain to describe the Equator and his Parallels, on the Polar or Orthognomonical Dyal.

IT shall suffice here to shew how the Parallels of the Signes may be described, because the Parallels of Declination and of the length of the day are described by like reason. And know that in the Polar plain because the Gnomon is Perpendi­cular to the plain, the same Gnomon shall serve both Hours and Azimuths; for the Hour-lines be Azimuths in this plain. Note [Page 191]also that the Sun is never Elevated above this plain more then he Declineth from the Equator, which at the most is 23 ½ degrees, and that if the height of Nodus be above a sixth part of the Semi­diameter of the plain, the ten first Almicantars will fall beside the plain. A sixth part therefore must serve, and that will give you all the Altitudes above to degrees, and the Parallels of the Signes whose Declination is more then 10.

Describe therefore the Almicantars here, as you were taught Chapter 27. for in the Hour lines you have already every 15 ^th Azimuth, and may draw more if you please.

Then know that the Almicantars here are the very Parallels of Declination, because the Equator is the Horizon of this plain,

and if you draw the Al­micantar 11.30 minutes, that shall be the Parallel of or ♍ because so much is the Suns Decli­nation and also his Al­titude above this plain, when he entreth those Signes the Almicantar for 20.13. minutes is the Parallel for ♊ and ♌ in like manner, and the Almicantar 23.30. mi­nutes the Parallel for ♋. And by like reason you may draw the Parallels of the South Signes on the South face.

CHAP. XXIX. How to inscribe the Equator and his Parallels, in the Equi­noctial or Parallelognomonical Dyal.

IF this plain Decline not, the Hour lines of your Country will serve you for they be also the Hour lines of the plain, and the Noon-line is Sub style if it do Decline, you shall draw in deleble lines the proper Dyal of the plain (by Chapter 23. which Declineth not.

And having here the Azimuths or Almicantars of the plain, [Page 192]drawn by Chapter 27. you shall observe upon your Planisphear at what Altitude or at what Azimuth the Parallels cut the seve­ral Hour lines, and where the like Altitude or Azimuth cuts the same Hour lines upon the plain, you shall make marks, and through those marks draw the Parallels which shall be all Coni­cal sections, except the Equator, which because he is a great Cir­cle, shall be a straight line on those plains; and in all other plains, except the Polar, where he is a Circle.

Now to find the Altitude and Azimuth which the Sun in such a Parallel hath at such or such an Hour. Take the Mater in the Meridional Projection; only for this time let the Limb be not the Meridian of your place, but the 6 Meridian or Hour-circle, and likewise Horizon of your plain, and let the Axtree line be the Meridian thereof, cutting it at Right angles. Now look where the first Hour-circle from the Limb (which is 7 a Clock) inter­secteth the Tropique of ♋, to that intersection you shall lay the Label (or rather one of the Semidiameters of the Reet, which toucheth neerer) and look how many degrees of the Limb lie be­tween

[Page 193]the Label and the Equator line so much is the Azimuth of that Hour from the East or West in the Tropique of Cancer: and the degree of the Label cut at the same time by the Tropique and the Hour line is the Altitude sought Do so for the second Hour circle, (which here is 8.) and so for the rest in order 9, 10, 11, 12. and you shall find that in the Tropique of Cancer in the Morning the Sun hath Azimuth from the East Northward, and Altitude as followeth,

In the Equator the Azimuth is always the same, full East or West, and so upon your plain he must needs be a straight line. The Alti­tudes in the Equator are 15, 30, 45, 60, 75, 90.

The Hours alike distant from the Meridian on both sides are alike, and so are the Parallels alike distant from the Equator a­like also.

When you have therefore gathered a Table out of your Pla­nisphear for the Morning Hours of the North Parallels, and of the Equator, (as I have done here in hast for the Equator, and Tropique of Cancer) you may by that Table prick down the Parallels upon one quarter of your Dyal, and by that also draw the rest; for as you may see upon your Planisphear, all the 4. quarters are alike.

Note that the Azimuths cut the Hour lines too Obliquely: it is best therefore to trust to the Almicantars, and so shall you have easier and surer work, though you meddle not with the A­zimuths at all.

CHAP. XXX. How to inscribe the Equator and his Parallels, in an Ob­lique or Scalenognomonical Dyal.

IF the plain neither Decline nor Recline, and so be a Verti­cal plain, the Hour lines of your Place will serve you for they be also the Hour lines of the plain, and the Noon line is the [Page 194]Sub-style. If it either Decline or Recline, or both Decline and Recline, you shall draw in deleble lines the proper Dyal of the plain by Chapter 23. and so this Dyal shall be reduced to a Ver­tical Dyal, and be as casily furnished with the Parallels as the Vertical: and when you have by the deleble Hour lines of the plain inscribed the Parallels, you may wipe out those Hour lines of the plain, and let the Hour lines of the place and the Parallels stand.

Having therefore drawn the Azimuths or Almicantars of your plain by Chapter 27, take your Planisphear in the Meridi­onal Projection, setting the Finitor to the Latitude of your plain; Then find your Equator and Parallels on the Mater, and where the several Hour lines intersect them above the Finitor, mark what Azimuth or rather what Almicantar passeth through these intersections; for in the same Azimuth and Almicantar shall the Parallels cut the Hour lines of the plain upon the plain.

Example. In the Vertical Dyal for our Latitude 52.15. mi­nutes, I set the Finitor to this Latitude, and going first to the Tro­pique of Cancer, I begin at the Limb, that is, at Noon, there I see the Azimuth full South the Altitude 61.15. minutes, at 1. a clock: Azimuth 27. Altitude 59. at 2. Azimuth 49 ½. Altitude 53 ⅓ at 3. Azimuth 67. Altitude 45 ½. &c. Therefore where I find the Noon line of my plain cut by the Almicantar 61 ¼, I make a prick, and in the Hour lines of 11 and of 1. where the A­zimuth 27 and the Almicantar 59 meet, I make pricks, and where the Azimuth 49 ½ and the Almicantar 53 ⅓ do meet up­on the Hour lines of 10.2. I make pricks, and so for the rest.

Lastly I draw with an even hand a crooked line without angles through those pricks, and that shall be the Parallel or Tropique of Cancer: and in like manner I put in all the other Parallels, and the Equator in the midst of them, though for the Equator you may draw him more speedily by striking a line through the Center of the Almicantars making Right angles with the Sub­style. And that may be a general Rule for the Equator in all Dyals which have a Substyle, and in the Polar Dyal where there is no Substyle, the Equator shall be a Circle, as before is shewn.

Note here that if your Dyal be great, and you have not points enough to govern you in the draught of the Conical sections you may draw half-hour-lines, and find points in them also, after the same manner.

[Page 195]

CHAP. XXXI. To do the same by the Hour-lines of the Place, although the Plain Decline or Recline.

IF you like not to draw the proper Dyal of the plain where it Declines or Reclines, because being useless in your Country it must be wiped out again, it shall suffice you to find the Hour lines of your Country upon the plain by Chapter 21, and 22 and in the posture your Planisphear hath in those Chapters to observe what Almicantars or Azimuths do cross those Hour lines at the several Hours in any Parallel, and thereby make pricks upon the Hour lines of your Place, as in the former Chapter you did upon the Hour lines of your plain; and by these pricks you may draw your Parallels as before.

Note that if you work this way, you shall find the Suns great­est Altitude to be in the Meridian of the plain or Substyle, and not in the Noon-line of your Place; whereat you must not wonder: so if the Substyle be about 9 in the Morning there you shall find the Sun at highest, and that his Altitude decreaseth from thence till he leaves the plain.

CHAP. XXXII. How to inscribe the Horizon of the Place, with his Azi­muths and Almicantars, in the Horizontal Dyal.

THe Nodus may be chosen in any part of the Gnomon, but with the eaution given Chapter 27. and a Perpen­dicular falling from the Nodus on the Sub-style shall touch the Center of the Azimuths and Almicantars of the plain, as hath been shewed Chapter 27.

Here you have no use of those Azimuths and Almicantars: but through the Center of them you shall draw an Horizontal line by Chapter 3. and that shall be Horizon.

Now if your plain Decline not from the Meridian, and so this Center fall upon the Noon line, you shall divide your Horizon both wayes from the Center, as you were taught to divide the Azimuthal lines by Tangents, Chapter 27. and shall number [Page 197]those divisions from the Center on both sides 5, 10, 15, 20. &c. and from the several points so made for 5, 10, 15, &c. In the Horizontal line let fall Perpendiculars or Vertical lines on the plain, and they shall be Azimuths of your Place.

But if your plain Decline you shall divide the Horizontal line thus. Draw a short Vertical line through the said Center down­wards, by Chapter 3. which shall be the Verticle of your plain.

In that Verticle set D E equal to the Radius of the Horizon, (that is, D N the height of Nodus) and upon the Center E de­scribe the arch G H or g h as wide as you please and the plain will bear; draw also through D the Horizontal line D F; then from the Center E strike the line E F f cutting the intersection of the Horizon with the Meridian of the Place at F and where the line E F cuts, either of the said arches. you shall begin to divide either of them into fives or tens of degrees numbring 5, 10, 15, &c. both wayes and laying a Ruler from E to those degrees in the arch you shall mark out thereby the same degrees in the Ho­rizontal line, and from these marks let fall Perpendiculars upon

the plain, which shall be the Azimuths desired.

For the Almicantars they will not be so handsome lines, but if [Page 198]you will have them, do thus. If the plain Decline not, set the Finitor to the Latitude of your Place, as Chapter 10. and if it De­cline, set the Finitor to the Latitude of your plain as Chapter 21. Then keeping your ey above the Horizon, and within the Tropiques, mark what Hour lines the 10 ^th Almicantar (for Ex­ample) cutteth, and what Azimuth there with him cutteth the same Hour lines also, and in the interfections of the same Azi­muths and Hour lines upon your plain you shall make marks, through which the tenth Almicantar shall be drawn: and so of the rest.

Note here That your Azimuths and Almicantars must not be drawn beyond the Tropiques, nor beyond the Horizon: neither must the Hour lines, if the Nodus be the end of the Gnomon.

The Scheam shews you how the Azimuths may be drawn on the Dyal of Chapter 12. and 21. Declining East 30 degrees.

CHAP. XXXIII. How by the help of this Furniture to place any moveable Dyal-Plain in his true Situation, and consequently to find the Meridian-line of the Place, without any other Instrument then the Dyal it self.

SEt the Dyal upon a level Table or Board, and turn it till the shadow of the Nodus touch the Suns Parallel, Azi­muth, or Almicantar, any or all of them: but the Pa­rallel shall best guide you, because that is most easily known by memory without Observation. And when the sha­dow of the Nodus toucheth the Suns Parallel it shews there the Hour also; and moreover it shews the Suns Altitude and Azi­muth for the same Time if the Azimuths and Almicantars also be drawn upon your Dyal.

But you shall note here, that the shadow of the Nodus may touch the Parallel at like distance from the Sub-style on both sides. Therefore if you be in doubt which is the true place of touching (as you may well doubt when the shadow cuts the Parallel near the Sub-style) you shall Observe a while whether the sh dow of the Radius be lengthning or shortning: If it shorten, the Sun is not come to the Sub-style, and so the earlyer [Page 199]Hour shewed is the true Hour; If it lengthen, the Sun is past the Sub-style, and the later Hour is the true Hour. And when the Dyal shews the true Hour, the Gnomon and the plains Parallel thereto do point North and South. And here you may see, that the further the Sun is from the Sub-style, the more easily is the Dyal placed.

Thus may you make a very commodious Polar Dyal, to stand in a chamber Window, and to remove from Window to Win­dow as the Sun goes, which shall find the Meridian line it self any where, in the 4. Summer and 4. Winter Moneths; and if you will make him a Limb, like the Limb of a Box-lid of a Cheese-fat, to receive the Parallels near the Equinoctial, which else fall beyond the plain, he shall serve for all the Year.

CHAP. XXXIV. How to make a Vertical Dyal upon the Ceeling of a Floor within Dores, where the Direct Beams of the Sun never come.

THe greatest part and as much as you shall use of the Vertical Dyal described Chapter 9. may by Reflection be turned upside down, and placed upon a Ceeling, but the Center will be in the Air without Dores.

A peece of a Looking-glass as broad as a Groat or Six-pence set level, or a Gally-pot of fair Water, which will set it self level, being placed upon the sole of the Window shall supply the use of the Nodus in the Gnomon, and the beams of the Sun being Reflected by this Glass or Water shall shew the Hours upon the Ceeling.

The Planisphear shall help you to make this Dyal two wayes. If the Window Decline not much from the South, you may make it most easily the First way. But if it Decline much, and so the lines fall much upon the partition Walls, or if you would adorn this Dyal with the Parallels or other Circles, you shall use the Second way.

The First way is this Draw a Meridian line upon the Floor. by Book 4.3. so that it may point upon the Perpendicular, which you shall imagine to fall from the Nodus upon the plain of the Floor prolonged. And this may be most easily done, if you [Page 200]hang a Plumb-line in the Window dnecuy over the Nodus of place of the Glais, for the shadow which that Plumb-line gives upon the Floor at Noon, is the Meridian line sought; and by a Ruler or a line stretched upon it you may prolong it as far as you shall need.

Then let a Plumb line fall from the Ceeling upon this Meridian line of the Floor, and behind it Northward or Southward, place your Ey, so that the Plumb-line may hide the Meridian line of the Floor from your Ey: then keeping your head steddy, cast you: Ey up to the Ceeling, and direct One to make two points at a good distance, in the line upon the Ceeling which the Plumb-line now covereth from your Ey, and by these points you shall draw a straight Meridian on the Ceeling.

Then having fastned one end of a Line at Nodus, let Another stretch this line up to the Meridian on the Ceeling, and let him move his hand nearer or further in the Meridian till you find by a Quadrant that this line pointeth up Northward as many de­grees as the Elevation of the Equator is in your Country, and then you shall cause him to make a point where the line toucheth [Page 201]the Meridian of the Cieling, and through that point you shall draw the Equinoctial line of your Dyal, cutting the said Meridian at Right angles. The length of the thred from the Nodus to the point in the Meridian where the Equinoctial cuts him, is Radius of the Equinoctial: to that Radius you shall find the Tangents of 15, 30, 45, 60, 75. as you found the Co-tangents Chapter 27. (knowing that the Co-tangents of 80, and 70. be the Tangents of 10, and 20, and so of the rest) and beginning in the Meridian make pricks in the Equinoctial line, at the end of the Tangent of 15. Eastward for 1. and Westward for 11. and at the end of the Tangent of 30. prick Eastward 2. and Westward 10. &c.

Then by Chapter 9. seek what angles the Hour lines of a Ver­tical Dyal make at the Center, which in our Latitude are 1.11.58. minutes, 2.24.32. minutes, 3.38.20. minutes, 4.53.52. minutes, 5.71.17. minutes, and with the Complements of these angles shall these Hour lines cross the Equinoctial: so the Hour line of 1. shall Incline to the Meridian on the South side the E­quinoctial line, and shall make his lesser angle with the Equi­noctial 78.02. minutes, and the rest as in the Figure.

The Second way is this. Fit a plain smooth Board about a foot Square to lie level from the fole of the Window inwards, and near the outer edge thereof make a Center in the board in the very place of Nodus, or a little under it, remembring that the Nodus or Center of the Glass must be set so much higher then this board, as the Center of your Quadrant is placed higher in the Projecting of the Dyal.

Upon that Center taken in the board describe as much of a Circle as you may with the Semidiameter of your Quadrant; which Circle shall be Horizon: Draw here from the Center to the Horizon inwards a Meridian line, by Book 4.3. and where it cuts the Horizon begin to graduate the Horizon into degrees of Azimuths both wayes, which you may speedily do, by transfer­ring the graduations of your Quadrant, or so much as you shall need, to this Horizon.

Next you must devise to make your Quadrant stand firm and upright upon one of his straight sides, (which I will call his foot for this time) and that you may thus do; Take a short peece of a Ruler or sinal Transom, and saw in one side of it a notch Per­perdicularly, in which notch you may stick fast or wedge the heel or the toe of your Quadrant, in such sort that his foot may come [Page 202]close to the board, and the other straight side or leg may stand Perpendicular upon it.

Those things prepared, put your Planisphear in the Meridional Projection, with the Finitor at your Latitude, and first observe there the Altitudes of the Sun in the Meridian, which in Latitude 52.15. minutes, you shall find in the Tropique of ♋ 61.15. mi­nutes, in the Equator 37.45. minutes, and in the Tropique of ♑ 14.15. minutes. Now having stuck a short needle in the Cen­ter of the Horizon, close to which you must alwayes keep the Center of your Quadrant, set the foot of your Quadrant in the Meridian line of the Board, and from the Center of your Qua­drant extend a thred by 14.15. minutes of Altitude straight on to the Cieling (the thred only touching the plain of the Quadrant and making no angle with it, but held Parallel) and where the thred thus extended touches the Cieling make a point, then the Qua­drant unmoved, extend the thred by 61.15. minutes of Altitude, and make another point as before, and between these two points draw a straight line, and that shall be your Meridian, and shall be long enough for your use: then extend the thred by 37.45. minutes of Altitude, and where it touches this Meridian cross the Meridian at Right angles with an infinite line, which shall be the Equator. Then seek upon your Planisphear for one a clock, and you shall find in the Tropique of ♑ the Suns Azimuth 14. and his Altitude 13.06. In the Tropique of ♋ his Azimuth 27½ and his Altitude 59.04. minutes: therefore setting the foot of the Quadrant in the Azimuth 14. from the Meridian Eastward, I extend the thred by 13.06. of Altitude, and make a prick in the Cieling: and again setting the foot of the Quadrant in Azimuth 27 ½. and extending the thred by 59.04 minutes of Altitude, I make another prick in the Cieling, and the straight line which I shall draw between these two pricks shall be all the Hourlines of One, and so of the rest. And if you be minded to have the o­ther Parallels drawn, you may find points for them as you have done for the Tropiques, and by those points draw them. And note that two points made in the Cieling for the same Hour line in any two Parailels, or in the Equator and any Parallel, shall suffice to direct the line, though it is best to take your points in the Tropiques, at the largest distance, as I have here done, if there be room enough on the Cieling.

But because it often happens that part of your Dyal falls beside [Page 203]the Cieling, and the plain of the Cieling and of the Walls is often interrupted, and made Irregular by Beams, Wal-plates, Cor­rishes, Wainscot, Chimney-peeces, and such like bodyes, I will [...]hew you one absolute device to carry on your Hour lines over all.

Extend the thred for any Hour line to the Tropique of Cancer [...]n the Cieling, as you where taught before, and fix it there, and extend another thred in like manner to the Tropique of Copricorn, where ever it shall happen, (as perhaps beyond the middle beam, or quite beyond the Cieling upon the Wall) and fix that thred al­so. Then place your Ey so behind these threds that one of them may cover the other, and at the same instant where the upper line (to your Sight or Imagination) cuts the Cieling, Beams, Wall, or any Regular or Irregular body, above the end of the lower line, there shall the Hour line pass from Tropique to Tro­pique: direct any By-stander to make marks as many as you shall need; and by these marks draw the Hour line according to your desire.

If the arch of the Horizon between the Tropiques be within view of your Window, you shall draw the same on the Wall to bound the Parallels, the Horizons Altitude you know is nothing, and therefore he will be a level line; and the Suns Azimuth when he riseth (commonly called Amplitude, and Ortive Lati­tude) is in Cancer 40.40. minutes East Northward, and in Ca­pricorn as much Southward; and these will be reflected to the contrary coasts on the Dyal.